There seems to be one character missing. But I gather that f(x) needs to satisfy
• [tex]x^3[/tex] divides [tex]f(x)[/tex]
• [tex](x-1)^3[/tex] divides [tex]f(x)^2[/tex]
I'll also assume f(x) is monic, meaning the coefficient of the leading term is 1, or
[tex]f(x) = x^5 + \cdots[/tex]
Since [tex]x^3[/tex] divides [tex]f(x)[/tex], and
[tex]f(x) = x^3 p(x)[/tex]
where [tex]p(x)[/tex] is degree-2, and we can write it as
[tex]f(x)=x^3 (x^2+ax+b)[/tex]
Now, we have
[tex]f(x)^2 = \left(x^3p(x)\right)^2 = x^6 p(x)^2[/tex]
so if [tex](x - 1)^3[/tex] divides [tex]f(x)^2[/tex], then [tex]p(x)[/tex] is degree-2, so [tex]p(x)^2[/tex] is degree-4, and we can write
[tex]p(x)^2 = (x-1)^3 q(x)[/tex]
where [tex]q(x)[/tex] is degree-1.
Expanding the left side gives
[tex]p(x)^2 = x^4 + 2ax^3 + (a^2+2b)x^2 + 2abx + b^2[/tex]
and dividing by [tex](x-1)^3[/tex] leaves no remainder. If we actually compute the quotient, we wind up with
[tex]\dfrac{p(x)^2}{(x-1)^3} = \underbrace{x + 2a + 3}_{q(x)} + \dfrac{(a^2+6a+2b+6)x^2 + (2ab-6a-8)x +2a+b^2+3}{(x-1)^3}[/tex]
If the remainder is supposed to be zero, then
[tex]\begin{cases}a^2+6a+2b+6 = 0 \\ 2ab-6a-8 = 0 \\ 2a+b^2+3 = 0\end{cases}[/tex]
Adding these equations together and grouping terms, we get
[tex](a^2+2ab+b^2) + (2a+2b) + (6-8+3) = 0 \\\\ (a+b)^2 + 2(a+b) + 1 = 0 \\\\ (a+b+1)^2 = 0 \implies a+b = -1[/tex]
Then [tex]b=-1-a[/tex], and you can solve for a and b by substituting this into any of the three equations above. For instance,
[tex]2a+(-1-a)^2 + 3 = 0 \\\\ a^2 + 4a + 4 = 0 \\\\ (a+2)^2 = 0 \implies a=-2 \implies b=1[/tex]
So, we end up with
[tex]p(x) = x^2 - 2x + 1 \\\\ \implies f(x) = x^3 (x^2 - 2x + 1) = \boxed{x^5-2x^4+x^3}[/tex]
AB = 6 and AC=13. Find BC.
Answer:
BC = 7
Step-by-step explanation:
AB + BC = AC
Substitute values for the line segments:
6 + BC = 13
Isolate line segment BC by subtracting both sides by 6:
6 - 6 + BC = 13 - 6
0 + BC = 13 - 6
BC = 7
Check your work:
6 + 7 = 13
13 = 13
Correct!
Given m||n, find the value of x. (3x+10) (6x-10)
Answer:
Use the given functions to set up and simplify
( 3x + 10 ) ( 6 x − 10 ) = 18 x^ 2 + 30 x − 100
solve giving brainlist if it right
F is the numerator of the fraction. To solve move g over to the opposite side by multiplying both sides by g:
F = h * g
Which of the following sets could be the sides of a right triangle?
{2, 3, √10 }
{3, 5,√34 }
{5, 8, 12}
I believe that it’s the last one because it makes the most sence but then again i could be wrong
if f(x) = 4x^2 - 3, then what is the remainder when f(x) is divided by x + 3?
Answer:
33
Step-by-step explanation:
x+3=0
x=-3
4(-3)²-3=(4×9)-3
36-3=33
If Kyle earns $12.50 per hour, how many hours did he work if he earned a total of $400. Set up an equation and solve. Let x = hours.
Answer:
32.
Step-by-step explanation:
Here is the equation I used. (x = hours)
$400 = $12.50x
To find how many hours he needs to work, divide both sides by 12.50 to get x by itself.
400/12.50 = 12.50x/12.50
Which equals:
32 = x
The base of a rectangle is 3cm more than the height. The perimeter is 26cm. Determine the length of the base in centimeters.
Answer:
Step-by-step explanation:
b = h+3
Perimeter = 2b+ 2h
= 2(h+3) +2h
= 4h+ 6
4h+6 = 26
h = 5 cm
b = h+3 = 8 cm
A popular amusement park has had199,792,458 visitors since it opened. Which
number, written in scientific notation, is the
best approximation of the number of visitors?
a) 0.2 x 107 visitors
b) 0.2 x 108 visitors
c) 2.0 x 107 visitors
d) 2.0 x 108 visitors
Answer:
d) 2.0 * 10⁸
Step-by-step explanation:
199,792,458 ≈ 200,000,000 = 2.0* 10⁸
giving 10 points image is attached
Answer:A
Step-by-step explanation:
4(4-w)=3(2w+2)
find w
Answer:
1
Step-by-step explanation:
4(4-w) = 3(2w+2)
Use distributive property:
16 - 4w = 6w + 6
+4w +4w
------------------------
16 = 10w + 6
-6 -6
-------------------------
10 = 10w
---- -----
10 10
1 = w
Hope this helped.
Hence, the required answer is 1
Solve for x: 3/2 + 1/2x =2x
[tex] \huge{ \boxed{ \bf{ \red{Problem:-}}}}[/tex]
Solve for x: 3/2 + 1/2x =2x[tex] \huge{ \boxed{ \bf{ \red{Solution:-}}}}[/tex]
[tex] \bold{ \frac{3}{2} + {\frac{1}{2} x = 2x}}[/tex]
[tex] \bold{ ⇒\frac{3}{2} + {\frac{x}{2} = 2x}} [/tex]
[tex] \bold{⇒\frac{3 + x}{2} = 2x}[/tex]
[tex] \bold{⇒3 + x = 2(2x)}[/tex]
[tex] \bold{⇒3 + x = 4x}[/tex]
[tex] \bold{⇒3 = 4x - x}[/tex]
[tex] \bold{⇒ \frac{3}{3} = x}[/tex]
[tex] \bold{ \therefore{x = 1}}[/tex]
Hope it's helpfulAnyone who is interested in construction, please help! Pweeeeseeee(;´༎ຶٹ༎ຶ`)
Answer:
I have ans in the picture form.
Step-by-step explanation:
Take it
Final grades are often computed using weighted averages. Some college professors use complicated weighted averages. Your college professor tells you he will give a 33%-67% split on grades between tests and homework where whichever average is higher will be the 67%.
What will your final grade be if your homework average is 85 and your test average is 76?
Answer:
≈ 82Step-by-step explanation:
Add 67% of 85 to 33% of 76:
85*0.67 + 76*0.33 = 82.03 ≈ 8267% of 85
[tex]\\ \sf\longmapsto \dfrac{67}{100}\times 85[/tex]
[tex]\\ \sf\longmapsto 85(0.67)[/tex]
[tex]\\ \sf\longmapsto 56.95[/tex]
33% of 76
[tex]\\ \sf\longmapsto \dfrac{33}{100}\times 76[/tex]
[tex]\\ \sf\longmapsto 76(0.33)[/tex]
[tex]\\ \sf\longmapsto 25.08[/tex]
Add both
[tex]\\ \sf\longmapsto 56.95+25.08[/tex]
[tex]\\ \sf\longmapsto 82.03[/tex]
Brooke spent 3/4 hours playing her flute each day for 7 days. How many hours did she play her flute in all
Answer:
5 and 1/4 hours or 5 hours and 15 minutes
Step-by-step explanation:
She played her flute 3/4 hours every day for 7 days, so to find the total number of hours we can multiply 3/4 by 7 to get 21/4. This is equal to 5 and 1/4 hours, or 5 hours and 15 minutes.
Answer: 5.25 hrs
Step-by-step explanation: 3/4 of an hour is 45 min. 45 min x 7 days is 315 min which is 5 hours and 15 min or 5.25 hrs
Which of the following best describes the graph below?
Answer:
the answer is c
Step-by-step explanation:
because it takes a non-negative number as input and returns the square root of that number as output.
Blake and josh are both saving money for a new bike the ratio of Blake,s saving to josh saving 3:1 if they have saved 400$, how much has each boy saved
Answer:
Bake = $300
Josh = $100
Step-by-step explanation:
3:1
3(00) / 1(00)
300 / 100
Chow,...!
please help me for my homework
1. (5x2-2x+3) + (x2+x+2)
Evaluate: – 10+26 (-7)||
A - 74
B 74
94
-94
please answer asap
how would I simplify by combining like terms. -5a - 4a + b
Answer:
Step-by-step explanation:
-5a-4a+b
-5a-4a are like terms
-5a-4a+b
-9a+b
please mark this answer as brainlist
what's the Answer ?
[tex] \sqrt{ \dfrac{225}{9} } [/tex]
Answer:
Solution given;
[tex] \sqrt{ \dfrac{225}{9} } [/tex]
[tex] \sqrt{ \dfrac{15²}{3²} } [/tex]
[tex] \sqrt{ (\dfrac{15}{3} )²} [/tex]
[tex]\dfrac{15}{3}[/tex]
=5
Answer:
First, simplify the content under root:
[tex]\sqrt{\frac{225}{9}} =\\\\\sqrt{25}}} \\[/tex]Now, use the following identity:
[tex]\sqrt{a^2} = |a|[/tex] = ± aLast, we get:
√25 = √5² = ± 5
Multiply. Express your answer in simplest form.
3 1/8 × 4/5
A 3 29/32
B 2 1/2
C 3 1/10
D 32/125
Answer:
B. [tex] 2 \frac{1}{2} [/tex]
Step-by-step explanation:
[tex]3 \frac{1}{8} \times \frac{4}{5} [/tex]
[tex] \frac{25}{8} \times \frac{4}{5} [/tex]
[tex] = \frac{5}{2} = 2 \frac{1}{2} [/tex]
what is the same as moving the decimal point 3 places to the left in a decimal number ?
1. Multiplying the number by 1,000
2. Dividing the number by 1,000
3. Multiplying the number by 100
4. Dividing the number by 100
When Beth last checked the clock it was 2:17 pm.
It is now 5:00 pm.
How much time has elapsed?
Answer:
The time is 2:43
Step-by-step explanation:
I did it in my head....
Which of the following statements must be true based on the diagram below? Select
all that apply. (Diagram is not to scale.)
L
K
O
M M
P
N
OP is a segment bisector.
OP is a perpendicular bisector.
OOP is an angle bisector.
O is the vertex of a pair of congruent angles in the diagram.
O is the vertex of a right angle.
Pis the midpoint of a segment in the diagram.
We have with a study the diagram of the diagram below and the explanations that follows the correct answers are
NONONOYESNOThe mid point of this segment can be derived by drawing diagonals across the 4 vertex LMNK.Generally
1)
OP is a segment bisector.
NO
OP is not a segment bisector because it those not divide segment LKMN in to two equal parts.
2)
OP is a perpendicular bisector.
NO
OP is not a perpendicular bisector because a perpendicular bisector is a bisector that cuts a segment into two equal parts and create an angle to the normal
3)
OP is an angle bisector
NO.
Because OP those not divide any angles into two equal parts
4)
O is the vertex of a pair of congruent angles in the diagram.
YES
Because O lines in between two angles of the same degree()
5)
O is the vertex of a right angle.
NO
Because O those not exist at a right angle
6)
The midpoint of a segment in the diagram
The mid point of this segment can be derived by drawing diagonals across the 4 vertex LMNK.
For more information on this visit
https://brainly.com/question/17581080
Solve for h.
2(h − 16) − 16 = –12
h =
Answer:
18
Step-by-step explanation:
2h - 32 - 16 = -12
2h - 48= -12
2h =-12+48
h=36\2
h = 18
Answer:
h = 18
Step-by-step explanation:
Given
2(h - 16) - 16 = - 12 ← distribute and simplify left side
2h - 32 - 16 = - 12
2h - 48 = - 12 ( add 48 to both sides )
2h = 36 ( divide both sides by 2 )
h = 18
i need some help!!!!!!!!!!!!!!!!!!!!!!!
Step-by-step explanation:
We are dealing with theoretical probability. This means that our probability of said outcome is
[tex] \frac{x}{y} [/tex]
where x is the outcome we want, and y is the total number of outcomes possible.
28. There is 3 triangles out of 6 polygons so the probability is 1/2.
29.There is 1 Pentagon so the probability is 1/6.
30. This is a complement to question 28. A complement is the inverse of the problem. A complement and its original add to 1 so the probability of both getting a triangle is 1/2.
31. There is 4 non quadraletrial so the probability is 2/3.
32. There is 3 figures that has more sides than three so 1/2 is the probability.
33.There is only multi right angles figures so 1/2 is the probability.
34. There is 30 days in April so the probability it's the 29th is
[tex] \frac{1}{30} [/tex]
35. There is 31 days in July and 15 days after the 16th. So the probability
is
[tex] \frac{15}{31} [/tex]
EASY MATH ALGEBRA 1 PLEASE HELP
Answer:
|-x|
Step-by-step explanation:
A reflection across the y-axis means the negative is on the inside: g(-x)
Find the solution to this equation.
x + 4 = 2x + 8
2
A
All real numbers
B.
No solution
Answer:
A. All real numbers.
Step-by-step explanation:
[tex]x + 4 = \frac{2x + 8}{2} \: \: multipling \: both \: sides \: by \: 2 \\ 2(x + 4) = 2x + 8 \\ 2x + 8 = 2x + 8[/tex]
When you end up with a true statement like this, it means that the solution to the equation is “all real numbers”. Any value for x that you can think of will make this equation true.
A soda company has sales of $35,135 million in 2010 and 45,954 million in 2014. Use the midpoint formula to estimate the sales in 2012. Assume the sales followed a linear pattern