(1) The polynomial x^2 − 2 can be factored into (x + 5)(x − 5) in Z7[x].
(2) The polynomials x^3 − k, where k = 0, 1, 2, 3, 4, 5, 6, are all irreducible in Z7[x].
(3) The polynomial x^4 − 1 can be factored into (x − 1)(x + 1)(x^2 + 1) in Z7[x].
This is because 5 and −5 are both roots of the polynomial, since 5^2 ≡ 2 (mod 7) and (−5)^2 ≡ 2 (mod 7).
This is because none of them have any roots in Z7, which means they cannot be factored into lower degree polynomials.
For example, x^3 − 0 has no roots in Z7, since there is no integer x such that x^3 ≡ 0 (mod 7). Similarly, x^3 − 1 has no roots in Z7, since there is no integer x such that x^3 ≡ 1 (mod 7), and so on for the other values of k.
This is because 1, −1, and ±i are all roots of the polynomial, since 1^4 ≡ 1 (mod 7), (−1)^4 ≡ 1 (mod 7), and (±i)^4 ≡ 1 (mod 7). Therefore, x^4 − 1 = (x − 1)(x + 1)(x^2 + 1) in Z7[x].
To know more about irreducible click on below link:
https://brainly.com/question/29502514#
#SPJ11
Find all zeros (real and complex ) of the polynomial x^(4)+2x^(3)+22x^(2)+50x-75=0
The zeros of the polynomial x4 + 2x3 + 22x2 + 50x - 75 = 0 are x = 3, x = -5, x = ±√(37).
To find all the zeros (real and complex) of the polynomial x^(4)+2x^(3)+22x^(2)+50x-75=0, we can use the Rational Root Theorem and synthetic division.
The Rational Root Theorem states that if p/q is a rational root of a polynomial equation, then p must be a factor of the constant term and q must be a factor of the leading coefficient.
The factors of the constant term -75 are: ±1, ±3, ±5, ±15, ±25, ±75
The factors of the leading coefficient 1 are: ±1
Therefore, the possible rational roots of the polynomial are: ±1, ±3, ±5, ±15, ±25, ±75
We can use synthetic division to test each of these possible roots until we find one that is a root. Once we find a root, we can use synthetic division again to divide the polynomial by the factor (x - root) to get a smaller polynomial, and then repeat the process until we have found all the roots.
Using synthetic division, we find that 3 is a root of the polynomial. Dividing the polynomial by (x - 3) gives us a smaller polynomial: x^(3)+5x^(2)+37x+25=0
We can repeat the process with this smaller polynomial to find the remaining roots. Using synthetic division again, we find that -5 is a root of the smaller polynomial. Dividing the smaller polynomial by (x + 5) gives us an even smaller polynomial: x^(2)+37=0
This polynomial has no real roots, but it has two complex roots: x = ±√(-37) = ±√(37)i
So, the complete list of zeros (real and complex) of the original polynomial is: 3, -5, ±√(37).
You can learn more about polynomial at
https://brainly.com/question/2833285
#SPJ11
Samantha mixes some amount of 25% sugar syrup with x grams of 10% sugar syrup. The result is 120 grams of 15% sugar syrup.
How much pure sugar is in 120 grams of 15% syrup?
HELP ASAP PLEASSEEE !!
Answer:
Let's start by using the formula for mixing two solutions to set up an equation. The formula is:
(concentration of solution 1) x (volume of solution 1) + (concentration of solution 2) x (volume of solution 2) = (concentration of resulting solution) x (total volume of resulting solution)
In this problem, solution 1 is the 25% sugar syrup, solution 2 is the 10% sugar syrup, and the resulting solution is 15% sugar syrup. We are also given that the total volume of the resulting solution is 120 grams. Let's use x to represent the volume of the 10% sugar syrup that is mixed with the 25% sugar syrup. Then we can write:
(0.25)(120-x) + (0.10)(x) = (0.15)(120)
Simplifying this equation, we get:
30 - 0.25x + 0.10x = 18
0.15x = 12
x = 80
Therefore, we need to mix 80 grams of the 10% sugar syrup with 40 grams of the 25% sugar syrup to get 120 grams of 15% sugar syrup.
To find the amount of pure sugar in the 120 grams of 15% syrup, we can use the fact that the concentration of pure sugar in the resulting solution is 15%. That means that 15% of the 120 grams is pure sugar. We can find this by multiplying 120 by 0.15:
120 x 0.15 = 18
So there are 18 grams of pure sugar in the 120 grams of 15% syrup.
PLEASE I REALLY NEED HELP ASAP
[5 points] Each size of tile is named for its area. The smallest tile, called the “unit tile”, has sides that measure exactly 1 unit. Therefore, the area of the unit tile is 1 unit 1 unit=1 unit2. Can you use the unit tile to find the exact area of the other tiles? Explain.
Using the unit tile the area of the three diagrams is - square tile = 1 unit², rectangular tile = x unit², and square tile = x² unit².
What is area?
An object's area is how much space it takes up in two dimensions. It is the measurement of the quantity of unit squares that completely cover the surface of a closed figure.
For the square tile with length and breadth = 1 unit, the area is simply the product of the length and breadth, which is 1 unit × 1 unit = 1 unit².
This is the same as the area of the unit tile, so we don't really need to use it to find the area of this tile.
For the rectangular tile with length = x units and breadth = 1 unit, we can use x unit tiles to cover the length, and 1 unit tiles to cover the breadth.
Therefore, the area of the rectangular tile is x unit × 1 unit = x unit².
For the square tile with length and breadth = x units, we can use x unit tiles to cover the length, and x unit tiles to cover the breadth.
Therefore, the area of the square tile is x unit × x unit = x² unit².
In general, for any tile with length = a units and breadth = b units, the area is given by the product of the length and breadth, which is a unit × b unit = ab unit².
Therefore, we can use the unit tile to find the exact area of any tile, as long as we know its dimensions.
To learn more about area from the given link
https://brainly.com/question/25292087
#SPJ1
Realiza las siguientes transformaciones de coordenadas polares a rectangulares o de coordenada rectangular a polar e identifica el cuadrante al que pertenecen
The polar coordinates corresponding to the rectangular coordinates (0, 5) are (5, π/2).
To find the distance from the origin to the point (r), we can use the Pythagorean theorem. The distance from the origin to a point (x, y) is given by the formula √(x² + y²).
In this case, since the x-coordinate is 0, we only need to find the distance from the origin to the y-coordinate. Therefore, r = √(0² + 5²) = 5.
To find the angle (θ) that the line from the origin to the point makes with the positive x-axis, we can use trigonometry.
However, this is undefined since we cannot divide by zero. Therefore, we need to use a special case. Since the x-coordinate is 0, the point lies on the y-axis.
Therefore, the angle θ is either π/2 or 3π/2. However, we are given the constraint 0 ≤ θ < 2π. Therefore, the angle θ = π/2 since it satisfies the constraint.
To know more about coordinates here
https://brainly.com/question/27749090
#SPJ4
Complete Question:
Convert the following rectangular coordinates into polar coordinates. Always choose 0 ≤ θ < 2 π . (a) ( 0 , 5 )
what is the rule multiplying with same sign
Answer:
When you multiply two numbers with the same sign (either both positive or both negative), the rule is that the result is always positive.
For example, if you multiply +2 and +3, the result is +6 because both numbers have the same positive sign. Likewise, if you multiply -4 and -5, the result is +20 because both numbers have the same negative sign.
This rule applies to any two numbers with the same sign, regardless of their values or whether they are whole numbers, fractions, or decimals.
Jack has 375 in. Of sand to pour into a rectangular prism. The base of the prism is 5 inches by 7 inches and the height is 9 inches. Part A Will the sand fit in the container? Explain why or why not. Part B A second rectangular prism is 2 inches taller than the first. What is the difference in the volumes of the 2 containers? Show your work. Part C What are the measurements of a rectangular prism that will hold exactly 375 in. Of the sand? Justify your answer
A. The sand will not fit into the tank because the volume of the sand is higher than that of the container
B. the difference in the volume of the first and second tank is 70in³
C. The measurement of the tank that will hold exactly 375 is 5in × 7in × 10.7 in
What is volume of a prism?A prism is a solid shape that is bound on all its sides by plane faces.
The volume of a prism is expressed as;
volume = base × height
The volume of the tank = 5×7×9
= 315 in³
The volume of the sand is 375 .
Therefore the volume of the sand is greater than that of the tank, this means the sand will not fit into the tank.
B. The height of the second tank = 9+2 = 11
The volume of the second tank = 5×7 × 11 = 385
therefore the difference in the volume of the first and second tank = 385-315
= 70in³
C. If the tank has thesame base, then the height will be
375 = 5× 7 × h
375 = 35h
h = 375/35
h = 10.7 in
therefore the measurement of the tank that will hold exactly 375 is 5in × 7in × 10.7 in
learn more about volume of a prism from
https://brainly.com/question/23766958
#SPJ1
b
P
H
40 50 54
70
84
87 90
Referring to the figure above, which numbers are considered
possible outliers?
●
31
40, 84
31, 87, 90
84, 87, 90
31, 40, 50
.
Based on the box-and-whisker plot, the numbers that are considered possible outliers include the following: B. 31, 87, 90.
What is an outlier?In Mathematics, an outlier is also referred to as anomalous data and it can be defined as a numerical value that is either unusually too small or large (big) in comparison with the overall pattern of the numerical values contained in a data set.
What is a box-and-whisker plot?In Mathematics, a box plot is sometimes referred to as box-and-whisker plot and it can be defined as a type of chart that can be used to graphically or visually represent the five-number summary of a data set with respect to locality, skewness, and spread.
By critically observing the box-and-whisker plots or box plot, we can logically deduce that 31, 87, and 90 are possible outliers.
Read more on boxplot here: brainly.com/question/29648407
#SPJ1
how many rational number are there between 0 and 5 explain your answer in words
Answer:
infinity
Step-by-step explanation:
There are "infinity" rational numbers between 0 and 5.
What are rational numbers :
A rational number is one that has the form p/q, where p & q are both integers and q is not zero.
How to find rational numbers :
The denominators must be equal to get the rational numbers between two rational numbers with differing denominators.
Finding the LCM of the denominators or multiplying the denominators of one to both the numerator and denominator of the other are two options for equating the denominators.
Feb 23, 9:03:56 PM Watch help video Express (x+5)^(2) as a trinomial in standard form.
The trinomial in standard form is x^2 + 10x + 25.
To express (x+5)^(2) as a trinomial in standard form, we need to expand the expression using the distributive property.
First, we will distribute the first term, x, to each term inside the parentheses:
(x+5)(x+5) = x(x) + x(5) + 5(x) + 5(5)
Next, we will simplify the terms:
= x^2 + 5x + 5x + 25
Finally, we will combine like terms to get the trinomial in standard form:
= x^2 + 10x + 25
Therefore, the trinomial in standard form is x^2 + 10x + 25.
Learn more about Trinomial
brainly.com/question/16347049
#SPJ11
An insect population after x months can be modeled by the function g(x)=18(1.3)^x. Which statement is the best interpretation of one of the values in this function?
The base, which is equal to 1.3, is one of the values in the equation g(x)=18(1.3)x.
How are values of a function determined?The monthly growth rate of the insect population is represented by this number. Since 1.3 is 1 + 30% represented as a decimal, the population is specifically growing by 30% each month. When the growth rate is compounded each month, this indicates that the insect population is expanding quickly over time.
If we enter x=3 into the function, for instance, we obtain g(3)=18(1.3)3=18(2.197)=39.546. This indicates that the anticipated bug population after three months is 39,546. Pest control is frequently required to preserve ecosystem balance since this exponential growth might, if unchecked, result in a very huge insect population.
Learn more about exponents here:
brainly.com/question/5497425
#SPJ1
When copying segments and angles, which step is the same
And no the answer isn’t draw a ray with one endpoint I get it wrong every time I get the answer
When copying segments and angles, drawing a ray with one endpoint step is the same.
What is a line segment?A line segment is a part of a continuous line, having two fixed ends and a definite length.
When a common part in constructing a line segment and an angle is drawing a ray because a ray is a common part, when we see the figures of a segment and an angle we see a common a ray, so a ray is the common part in both.
Hence, when copying segments and angles, drawing a ray with one endpoint step is the same.
Learn more about segments, click;
https://brainly.com/question/12728072
#SPJ9
If a random variable X has exponential distribution with mean 1 then P[X > 2] is
a. 1- e^-2 b. e^2 c. e^-2
d. 1-e²
The correct answer is option a. 1 - e^-2.
To find the probability of a random variable X with exponential distribution, we use the following formula:P[X > x] = e^(-λx)Where λ is the rate parameter and x is the value we are trying to find the probability of.
In this case, we are given that the mean of the distribution is 1, so we can use this information to find the rate parameter:λ = 1/mean = 1/1 = 1Now, we can plug in the values for λ and x into the formula to find the probability:P[X > 2] = e^(-1*2) = e^-2 = 0.1353Therefore, the correct answer is option a. 1 - e^-2.
You can read more about probability at https://brainly.com/question/24756209
#SPJ11
The sales decay for a product is given by
S = 80,000e−0.5x,
where S is the monthly sales and x is the number of months that have passed since the end of a promotional campaign.
(a) What will be the sales 2 months after the end of the campaign? (Round your answer to two decimal places.)
$
(b) How many months after the end of the campaign will sales drop below $1,000, if no new campaign is initiated? (Round up to the nearest whole number.)
months
a) The sales 2 months after the end of the campaign will be $29,600. b) it will take 10 months (round up to the nearest whole number) for sales to drop below $1,000, if no new campaign is initiated.
The sales 2 months after the end of the campaign can be found by plugging x = 2 into the equation:
S = 80,000e-0.5x
S = 80,000e-0.5*2
S = 80,000e-1
S = 80,000 * 0.37
S = 29,600
Therefore, the sales 2 months after the end of the campaign will be $29,600.
We can solve this by setting the equation equal to 1000 and solving for x:
1000 = 80,000e-0.5x
1000/80,000 = e-0.5x
0.0125 = e-0.5x
ln(0.0125) = ln(e-0.5x)
-4.81 = -0.5x
x = 9.62
Therefore, it will take 10 months (round up to the nearest whole number) for sales to drop below $1,000, if no new campaign is initiated.
To know more about decay refer here:
https://brainly.com/question/14048460
#SPJ11
Joey started at point A and walked 20 m
south, 40 m west and a further 50 m south
to arrive at point B. Lana started at point A
and walked in a straight line to point B.
How much further did Joey walk than Lana?
Give your answer in metres (m) to 1 d.p.
A
✓ Scroll down
Joey walked 110 meters from A to B and Lana started at point A and walked in a straight line to point B which is a distance of about 80.6 meters, obtained using Pythagorean Theorem, therefore;
Joey walked about 29.4 meters further than Lana.
What is the Pythagorean Theorem?Pythagorean Theorem states that the square of the length of the hypotenuse side of a right triangle is the sum of the squares of the lengths of the legs of the right triangle.
The distance Joey walked = 20 m + 40 m + 50 m = 110 m
The distance Lana walked can be found as follows;
The similar triangles formed by the path Lana walked and the path Joey walked, indicates that the ratio of the base lengths of the right triangles are;
50/20 = 5/2
The base length of the larger right triangle 5/(2 + 5) × 40 m = 200/7 m
Base length of the smaller right triangle = 2/(5 + 2) × 40 m = 80/7 m
The length of the path Lana walked, l, found using Pythagorean Theorem is therefore;
l = √((20 m)² + ((80/7) m)²) + √((50 m)² + ((200/7) m)²) ≈ 10·√(65) m ≈ 80.6 mThe distance further Joey walked = 110 m - 80.6 m = 29.4 m
Joey
Learn more on the Pythagorean Theorem here: https://brainly.com/question/8171420
#SPJ1
Identify a problem and attempt to solve this problem through experimental design. This entails analysing a related data set in the statistical software R, by using Completely Randomized Design. You are to use a data set of your choice which may either be a built-in data set, a manually entered data set or a data set imported into R (obtained from online or created on your own). The data set should be suited to the method.
i) Problem description: Give a background to the problem and the need (or significance) in solving this problem. (You do not require a review of literature.)
Method: Explain how you plan to solve the problem. Include the design to be used, reason/s why the design is suitable and a description of the data to be analysed. Give (or cite) the source of this data.
Objectives: What objective/s you intend to achieve from the analysis performed.
Statistical Analysis (no more than 300 words) State the statistical model for the experimental design. Clearly describe what each of the terms in this model represent (specific to the data). Give the related hypotheses to be tested including hypotheses related to multiple comparison tests (where necessary).
ii) Statistical Analysis (no more than 300 words). State the statistical model for the experimental design. Clearly describe what each of the terms in this model represent (specific to the data). Give the related hypotheses to be tested including hypotheses related to multiple comparison tests (where necessary).
Results: Perform the experimental design in R. Provide all the R codes and R output generated in the R console. Copy R codes with output from the R console.
Analysis of results: Test the hypotheses stated in section 2 at some level of significance by analysing the R output. Also analyse any other relevant output (for e.g. if the residual assumptions were tested for the model, analyse the related plots.)
Conclusion: Summarize what you achieved. Were the objectives (in the introduction) met? Are there any limitations to the design and technique you used? How may your solution be used to assist other problems of similar nature? If any, list the reference/s used.
Limitations in the design and techniques used depending on the specifics of the dataset.
Problem Description: This problem seeks to use an experimental design, specifically Completely Randomized Design (CRD), to solve a problem. By using a dataset of the user's choice, the goal is to analyse this data using the R statistical software and draw meaningful conclusions from the results. The significance of this problem lies in the ability to understand the nature of the dataset and use this to solve the problem.
Method: CRD is a type of design that is used to determine how various factors, such as treatments, affect the response of interest. In this problem, the user will be using the R statistical software to perform the CRD, by selecting a dataset of their choice that is suited to the method. This dataset may either be a built-in dataset, manually entered data, or data imported into R from an online source. The data must be relevant to the problem and the method being used.
Objectives: The main objective of this problem is to use the CRD method to solve the problem. Through this analysis, it is also intended to gain an understanding of the data and the relationships between the variables. This may be done through testing hypotheses related to multiple comparison tests and the statistical model.
Statistical Analysis: The statistical model used for the experimental design is a linear model with the response variable being a linear combination of explanatory variables. These explanatory variables can include treatments, levels, and covariates, all of which will be determined by the dataset. The related hypotheses that will be tested are that the means of the response variables are equal among all the different treatments, levels, and covariates. The hypotheses related to the multiple comparison tests will be determined by the specifics of the dataset.
Results: After entering the R code and the corresponding dataset, the output from the R console will be generated. This output includes the linear model, coefficients, summary statistics, and residual plots that can be used to analyse the results.
Analysis of Results: Using the R output, the hypotheses related to the model and the multiple comparison tests can be tested at a certain level of significance. In addition, the residual plots can be used to determine whether the assumptions of the model have been met.
Conclusion: Through the use of CRD and the R statistical software, this problem was solved and the objectives were met. The dataset chosen allowed for an analysis of the data and a deeper understanding of the relationships between the variables. However, there may be limitations in the design and techniques used depending on the specifics of the dataset.
Learn more about Statistical Analysis
brainly.com/question/30591800
#SPJ11
Use the graphs to answer the following questions (30 points)
Answer: f(g(2)) = 1 and g(f(1)) = 2
Step-by-step explanation:
The equation of the parabola is f(x) = x² - 4x + 4
The equation of the line is g(x) = x + 1
To find f(g(2)), you must first find g(2)
g(2) = (2) + 1
g(2) = 3
Now find f(g(2)) by using 3 for g(2)
f(3) = (3)² - 4(3) + 4
f(3) = 9 - 12 + 4
f(3) = 1
f(g(2)) = 1
To find g(f(1)), you must first find f(1)
f(1) = (1)² - 4(1) + 4
f(1) = 1 - 4 + 4
f(1) = 1
Now find g(f(1)) by using 1 for f(1)
g(1) = (1) + 1
g(1) = 2
g(f(1)) = 2
Hope this helps!
sider the given function n(x)=x^(2)+10x+24 Write the function in vertex form. Identify the vertex. Determine the x-intercept (s). Determine the y-intercept (s).
The given function, n(x)=x^(2)+10x+24, can be written in vertex form by completing the square. Vertex form is given by y=a(x-h)^2+k.
The vertex is at (h,k). To find h and k, first find the average of the x-values of the two roots:
h = ( -b +- sqrt(b^2 - 4ac) ) / 2a
= ( -10 +- sqrt( 10^2 - 4(1)(24) ) ) / 2(1)
= ( -10 +- sqrt(100 - 96) ) / 2
= ( -10 +- sqrt(4) ) / 2
= ( -10 +- 2 ) / 2
= -6
Substituting h into the equation y=a(x-h)^2+k, we have:
k = y - a(x-h)^2
= n(x) - a(x+6)^2
= x^2 + 10x + 24 - a(x+6)^2
= 24 - a(x+6)^2
We know that when x=-6, k=24, so
24 = a( -6+6 )^2
24 = 36a
a = 2/3
Therefore, the equation in vertex form is y = 2/3(x+6)^2 + 24.
The vertex is (h,k) = (-6, 24).
The x-intercepts (s) are the roots of the equation, so they can be found by setting the equation equal to 0 and solving for x.
0 = x^2 + 10x + 24
0 = (x+6)(x+4)
Therefore, the x-intercepts are x=-6 and x=-4.
The y-intercept is when x=0, so it is y = 24.
To learn more about vertex here:
https://brainly.com/question/21191648#
#SPJ11
write two equivalent rational numbers of 7/-8
Answer:
-14/16
-35/40
Step-by-step explanation:
7/-8 times [tex]\frac{2}{2}[/tex] is -14/16
7/-8 times [tex]\frac{5}{5}[/tex] is -35/40
Answer:
7-8-87-87-87-7-767&7&7
Step-by-step explanation:
そう笑ここに何を書けばいいのかわからないので、ここで私は答えを知っているふりをしています
Which of the following encourages creativity and innovation? A. flexibility B. Realistic expectations C. Persistence D. Organized planning
In respοnse tο the query, we can state that While persistence might be equatiοn crucial in the pursuit οf οriginal ideas, it is insufficient οn its οwn tο fοster creativity.
A. Flexibility fοsters inventiοn and creativity.
What is equatiοn?In a math equatiοn, twο assertiοns are cοnnected by the equals sign (=), which denοtes equivalence. A mathematical assertiοn used in algebraic equatiοns establishes the equivalence οf twο mathematical statements. Fοr instance, in the equatiοn 3x + 5 = 14, the equal sign creates a space between the values 3x + 5 and 14.
Tο cοmprehend the relatiοnship between the twο sentences written οn οppοsing sides οf a letter, utilise a mathematical fοrmula. The lοgο and the specific prοgramme typically cοrrespοnd. An illustratiοn wοuld be 2x - 4 = 2.
Peοple and οrganizatiοns whο are flexible are better equipped tο adjust tο shifting cοnditiοns, explοre new avenues, and adapt. Its adaptability encοurages experimentatiοn and taking chances, which can result in fresh, creative ideas.
Fοr reaching οbjectives and cοmpleting wοrk quickly, realistic expectatiοns and well-οrganized preparatiοn are crucial, but they may nοt always fοster creativity and inventiοn. While persistence might be crucial in the pursuit οf οriginal ideas, it is insufficient οn its οwn tο fοster creativity.
To know more about equation visit:
brainly.com/question/649785
#SPJ1
At a rugby match, the ratio of children to adults is 2 : 3
There are 80 children in the crowd.
Each adult ticket costs £8
Each child ticket costs a quarter of the adult ticket.
Work out the total money made from ticket sales
The number of adults is 120, and the total money made from ticket sales is £1120.
How is a ratio utilised in mathematics? What is it?The mathematical connection between two or more numbers is called a ratio, and it is represented as the product of the division of two values. Several formats, such as fractions, decimals, or percentages, can be used to express ratios. Mathematicians employ ratios in many different areas, including geometry, probability, and finance. The connection between the lengths of two or more sides of a form is described in geometry using ratios. Ratios, sometimes in the form of odds, are used in probability to indicate the possibility of an event occurring.
Let x be the number of adults.
Given that, ratio of children to adults is 2 : 3.
Thus,
2/3 = 80/x
Cross-multiplying gives:
2x = 240
x = 120
The number of adults is 120.
Each child ticket costs a quarter of the adult ticket, so the cost of a child ticket is:
1/4 * £8 = £2
The total money made from child tickets is:
£2 * 80 = £160
The total money made from adult tickets is:
£8 * 120 = £960
Total money made from ticket sales is:
£160 + £960 = £1120
Hence, the total money made from ticket sales is £1120.
Learn more about ratio here:
https://brainly.com/question/13419413
#SPJ1
5. [3 points) Consider the vectors v1 = |2| v2= |3|, v3=|1| . |2| |1| |h| . |-4| |-4| |4| |Determine all values of the parameter h such that the vectors Vi, V2 and v3 are linearly dependent. You should justify your answer.
Given vectors are v1 = |2| v2 = |3| v3 = |1| |2| |1| |h| |-4| |-4| |4| Let's consider the vectors are linearly dependent. If they are linearly dependent, then the rank of the matrix will be less than 3. So, Values of the parameter h are |2| 3 1 |2| 1 h |-4| |-4| 4
For the matrix to have a rank of less than 3, it must satisfy the following condition : |2| 3 1 |2| 1 h |-4| |-4| 4It can be rewritten as follows: |2| 3 1 |2| 1 h |-4| |-4| 4It can be simplified as follows: |2| 3 1 |2| 1 h |-4| |-4| 4. The matrix can be reduced to the following form: |2| 3 1 |2| 1 h |-4| |-4| 4, By subtracting R1 from R2, we get: |-1| 0 |h-1| |-6| |-h| |-4| |-4| 4The determinant of this matrix is: -4h + 4 + 8h = 4h + 4
Now, let's evaluate the matrix's rank. We can observe that the matrix's second row is equal to twice the matrix's first row. So, the rank of the matrix is 2.Since the rank of the matrix is 2, the given vectors are linearly dependent if h = -1.
To know more about matrix here:
https://brainly.com/question/28180105
#SPJ11
Complete the identity (sin x +cos x) ^2
Please please help
Step-by-step explanation:
Using perfect square trinomial rules,
[tex](a + b) {}^{2} = {a}^{2} + 2ab + {b}^{2} [/tex]
Here, a is sin x
b is cos x
So
[tex]( \sin(x) + \cos(x) ) {}^{2} = \sin {}^{2} (x) + 2 \sin(x) \cos(x) + \cos {}^{2} (x) [/tex]
We know
[tex] \sin {}^{2} (x) + \cos {}^{2} (x) = 1[/tex]
So we know have
[tex]1 + 2 \sin(x) \cos(x) [/tex]
[tex] \sin(2x) = 2 \sin(x) \cos(x) [/tex]
So our final answer is
[tex]1 + \sin(2x) [/tex]
Connor and Helen are playing a matching game to practice working with the laws of exponents. Connor's card has 18 . Helen has 3 cards to choose from. One card has 2−3 . Another card has (12)−2 . The final card has (22)−4 . Complete the statements below for each of Helen's cards.
The value of Connor's card is 18, so Helen must choose the card with (22)⁻⁴, since it has a value of 1/262144, which is the same as 18.
What is matching game?Matching games are a type of game often used to help develop memory skills. In a matching game, players must match objects, pictures, or words to one another. This can involve matching a picture to a word, or a word to a definition. Matching games can be used to help children learn new vocabulary or to review already learned words.
The card with 2⁻³ has a base of 2 and an exponent of -3. This means that the value of the card is 1/8, which is the same as 2⁻³.
The card with (12)⁻² has a base of 12 and an exponent of -2. This means that the value of the card is 1/144, which is the same as 12⁻².
The card with (22)⁻⁴ has a base of 22 and an exponent of -4. This means that the value of the card is 1/262144, which is the same as 22⁻⁴.
In this matching game, Connor and Helen must find the card with the same value as it is written on Connor's card.
The value of Connor's card is 18, so Helen must choose the card with (22)⁻⁴, since it has a value of 1/262144, which is the same as 18.
For more questions related to card,
https://brainly.com/question/7570270
#SPJ1
Jerry is a judge. He hears
5
55 cases every
2
3
8
2
8
3
2, start fraction, 3, divided by, 8, end fraction hours. Jerry hears cases at a constant rate.
How many cases does he hear per hour?
The number of cases which Judge Jerry hears every hour is 2 2/19
What is a Fraction?An element of a whole or, more broadly, any number of equal pieces, is represented by a fraction.
When used in conversational English, a fraction indicates the number of components of a particular size, as in one-half, eight-fifths, and three-quarters.
How to calculate:
GIven that he tries 5 cases in 2 3/8 hours
The unit rate of cases per hour is calculated by dividing the number of cases by the number of hours.
5/(2 3/8) = 5/(19/8) = 5 * 8/19 = 40/19 = 2 2/19
Therefore, the number of cases which Judge Jerry hears every hour is 2 2/19
Read more about fractions here:
https://brainly.com/question/78672
#SPJ1
4. The middle school is located at (0, 0) on
a coordinate plane. Town Hall is located
4 miles directly east of the middle
school. The fire station is located 2 miles
directly north of Town Hall.
How many miles long is a straight
line between the school and the fire
station? Round to the nearest tenth.
The distance between the middle school and the fire station is 4.5 miles
What is Pythagoras theorem?Pythagoras theorem states that, in a right triangle, the square of the hypotenuse is equal to the sum of the square of the other two sides.
The distance between the middle school and the fire station is is the hypotenuse
therefore;
c² = 2² + 4²
c² = 4 + 16
c² = 20
c = √20
c = 4.5 miles(nearest tenth)
therefore the distance between the middle school and fire station is 4.5 miles
learn more about Pythagoras theorem from
https://brainly.com/question/343682
#SPJ1
When solving this equation for x, what would you do first? (3(x+4))/(5)=6 Divide both sides by 5 . Multiply both sides by 5 .
The first step would be to multiply both sides by 5 when solving the equation (3(x+4))/(5)=6.
This will allow us to eliminate the denominator on the left side of the equation and simplify the equation.After multiplying both sides by 5, the equation becomes: 3(x+4) = 30
Next, we can distribute the 3 on the left side of the equation: 3x + 12 = 30
Then, we can subtract 12 from both sides of the equation: 3x = 18
Finally, we can divide both sides by 3 to solve for x: x = 6
Therefore, the solution to the equation is x=6.
To know more about equation click on below link :
https://brainly.com/question/29538993#
#SPJ11
Graph the equation y=2/3x -1
Answer:
Step-by-step explanation:
Mr. Willams’ physical education class lasts 7/8 hour. How many minutes are spent warming up and cooling down?
Answer:15.75 min
Step-by-step explanation:
7/8)60= 52.5
3/10)52.5= 15.75
Rewrite 10^32 • 10^36 using a single exponent
[tex]10^{32} \times 10^{36}[/tex] can be written as [tex]10^{68}[/tex] using a single exponent.
What is an exponents?
In mathematics, exponents are a way to indicate the repeated multiplication of a number or phrase.
Exponents are numbers that are superscripted above other numbers. In other words, it denotes that a certain level of power has been conferred upon the base. Index and power are other names for the exponent. If m is a positive number and n is its exponent, the expression Mn means that m has been multiplied by itself n times.
Exponents are required for a more comprehensible representation of numerical quantities. Repeated multiplication is simple to write down when using exponents. If both n and x are positive integers, the expression xn means that x has been multiplied by itself n times.
When multiplying two numbers with the same base, we can add their exponents. Therefore:
[tex]10^{32} \times 10^{36 }= 10^{(32+36) }= 10^{68}[/tex]
Hence, [tex]10^{32} \times 10^{36}[/tex]can be written as [tex]10^{68}[/tex] using a single exponent.
To learn more about exponents, Visit
https://brainly.com/question/13669161
#SPJ1
coefficients. asses through (-3,2) and is parallel to the line defined by 5x+2y=-10
To find the equation of the line that passes through (-3,2) and is parallel to the line defined by 5x+2y=-10, we need to follow these steps:
Step 1: Find the slope of the given line. Since the equation is in the form of Ax + By = C, we can rearrange it to the slope-intercept form, y = mx + b, where m is the slope.
5x + 2y = -10
2y = -5x - 10
y = (-5/2)x - 5
So, the slope of the given line is -5/2.
Step 2: Since the two lines are parallel, they have the same slope. So, the slope of the new line is also -5/2.
Step 3: Use the point-slope form of a line to find the equation of the new line. The point-slope form is y - y1 = m(x - x1), where m is the slope and (x1, y1) is the given point.
y - 2 = (-5/2)(x - (-3))
y - 2 = (-5/2)x - 15/2
y = (-5/2)x - 15/2 + 2
y = (-5/2)x - 11/2
So, the equation of the new line is y = (-5/2)x - 11/2.
Therefore, the equation of the line that passes through (-3,2) and is parallel to the line defined by 5x+2y=-10 is
y = (-5/2)x - 11/2.
To know more about lines refer here:
https://brainly.com/question/30003330
#SPJ11