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]
An expression is shown below.
6ab^2 + 9a^2b
Which of the following shows an equivalent expression?
Answer:
C. 3ab ( 2b + 3a )
Step-by-step explanation:
substitution
Hi! I'm happy to help!
To first solve this, Let's see if we can simplify our expressions using the order of operations: our original expression cannot be simplified. The first expression cannot be simplified. The second expression we can multiply the outside by the parenthesis by the inside, same with the third and fourth.
(make sure everything outside of the parenthesis is being multiplied by each part inside of the parenthesis)
9a²b²
9a²b²
3ab(3b+2a)
ab(9b+6a)
b(9ab+6a²)
9ab²+6a²b
3ab(2b+3a)
ab(6b+9a)
b(6ab+9a²)
6ab²+9a²b
3a²b²(2b+3a)
a²b²(6b+9a)
b²(6a²b+9a³)
6a²b³+9a³b²
Now that we have all expressions fully simplified, let's compare to see which one matches.
Our first expression is nowhere close to what our original expression is, 1 is incorrect.
Our second expression is close, but it has the 9 and the 6 flipped, even if we rearrange the expression like this, 6a²b+9ab², it still doesn't match, so 2 is also incorrect.
Our third expression has the correct numbers, variables, placement, and exponents, so 3 is correct.
Our fourth expression has incorrect exponents, the expression made every variable multiplied by ab an extra time, so 4 is incorrect.
In summary, you should pick number 3, because it is equivalent.
I hope this was helpful, keep learning! :D
What is the length of YZ?
Select the best answer from the choices provided.
A.4
B.4.58
C. 5.29
D.10
Answer: C
Hello,
Step-by-step explanation:
WZ=7-3=4
YW²=XW*WZ=3*4=12 ==> WY=2√3
In the right triangle YWZ, using the Pythagorean's theorem:
YZ²=WY²+WZ²=12+4²=28
YZ=2√7≅5.29150....
Answer C
A swimming pool holds liters of water. The pool has two drainage pipes. When the pool is completely full, the first pipe alone can empty it in minutes, and the second pipe alone can empty it in minutes. When both pipes are draining together, how long does it take them to empty the pool
Answer:Pipe 1 is emptying the pool at 600,000/200 = 3000 liters per minute.
The second pipe empties the pool at 600,000/120 =5000 litres per minute. With both pipes the pool will be emptied at 8000 litres per minute. It will therefore take 600,000/8000 = 75 minutes to empty the pool of water when both pipes are draining together.
Step-by-step explanation:
You are making spaghetti for dinner. You have the choice of marinara, alfredo, or meat sauce, 4 different shaped noodles, and 2 kinds of cheese. How many different spaghetti dinners can you make?
Answer:
DCC
Step-by-step explanation:
CDDCDC
Different type of spaghetti dinners is 8
What is combination ?
"The number of possible arrangements in a collection of items where the order does not matter."
For 4 different shaped of noodles = ⁴c₁
⁴c₁ = 4
For two kinds of cheese = ²c₁
²c₁ = 2
Now , multiplying these 4×2 = 8
Hence, 8 type of spaghetti dinners
For more about combination
https://brainly.com/question/21306575
#SPJ2
What is the range of the function shown on the graph above?
A. -8 y -3
B. -8 y -3
C
Answer:
i got u c
Step-by-step eplanation:
Solve for b.
2(b − 8) + –11 = 3
b =
Answer:
b = 15
Step-by-step explanation:
2 (b - 8) + -11 = 3
2b - 16 + -11 = 3
2b - 16 + -11 = 3
2b - 27 = 3
2b = 3 + 27
2b = 30
b = 30 ÷ 2
b = 15
Answer:
[tex] = > 2(b - 8) + - 11 = 3[/tex]
[tex] = > 2b - 16 + - 11 = 3[/tex]
[tex] = > 2b - 27= 3[/tex]
[tex] = > 2b = 27 + 3[/tex]
[tex] = > 2b = 30[/tex]
[tex] = > b = \frac{30}{2} [/tex]
[tex]=>b=15[/tex]
pls answer my question
Answer:
hope this helps you
have a great day
Find the 11th term of the arithmetic sequence
2x - 2, 5x – 9, 8x – 16,...
Answer:32x -72
Step-by-step explanation:
Need help with this question. Will give brainliest for correct answer.
Answer: the second to last
Step-by-step explanation: because there’s no multipcation in the first box and if you look at the rest it wou make sense for it to be the second to last
(3,-2)
– 2-
→
(1,0)
how many times am i moving ?
Answer:
Step-by-step explanation:
What is 898995x10007766767
Answer: 8.9969323e+15
if n=4 and m=3 find 2n-3m
Answer:
hope it helps you........
Answer:
2n-3m
=2(4)-3(3)
=8 - 9
= -1
f(x)= 2/3x +3; Find when f(x)=-5
Hi ;-)
f(x) = ²/₃x + 3 ∧ f(x) = -5
²/₃x + 3 = -5
²/₃x = -5 - 3
²/₃x = -8 / * ³/₂
x = -12
Write an equation that represents the line.
Use exact numbers.
Answer:
y = 3/4m + 2
Step-by-step explanation:
Distance from point 1 to point 2 is up 3, over 4
Point 1 intercepts (0, 2).
Ann drove 378 kilometers this weekend. She drove 49 of the distance on Saturday. How many kilometers did she drive on Saturday?
Answer:
185.22 km
Step-by-step explanation:
Assuming that you forgot the percentage sign In the question. Ann drove 185.22 km on Saturday.
Andrew bought four bottles of lemonade and two bottles of water for €2.45. From the same shop Michael
bought two bottles of lemonade and four bottles of water for €2.35. What is the total cost of one bottle of
lemonade and one bottle of water?
Answer:
0.8 euro
Step-by-step explanation:
Suppose the price of bottle of lemonade is x, when the bottle of water costs y.
Four bottles of lemonade cost 4x, two bottles of water cost 2y
4x+2y=2.45
Two bottles of lemonade cost 2x, four bottles of water cost 4y
2x+4y= 2.35
These equations( 4x+2y=2.45 and 2x+4y= 2.35) are simultaneous, so
multiply the first one by -2
-8x-4y=-4.9
-8x-4y+2x+4y= -4.9+2.35
-6x=-2.55
x=0.425 euro - the cost of bottle of lemonade
y=(2.45-4*0.425)/2
y=0.375 euro - the cost of bottle of water
0.425+0.375= 0.8 euro - the total cost of one bottle of
lemonade and one bottle of water
Answer:
price of lenonade bottle is 0.425
price of water bottle is 0.375
Step-by-step explanation:
In explaination, 'x' is the value of lemonade bottle and 'y' is the value of water bottle.
All explaination is in picture below.
Show all work to solve the equation for x. If a solution is extraneous, be sure to identify it in your final answer.
square root of the quantity x minus 3 end quantity plus 5 equals x
x=7 is the solution for the given equation
x=4 is the extraneous solution .
Given :
[tex]\sqrt{x-3} +5=x[/tex]
Explanation :
To solve the given equation for x , we need to remove the square root first
Subtract 5 on both sides
[tex]\sqrt{x-3}+5=x\\ \sqrt{x-3}=x-5\\\\[/tex]
Take square on both sides to remove square root
[tex]x-3=(x-5)^2\\x-3=x^{2} -10x+25\\[/tex]
Subtract x and add 3 on both sides
[tex]0=x^{2} -11x+28\\0=(x-7)(x-4)\\[/tex]
Set the factors =0 and solve for x
[tex]x-7 =0\\x=7\\\\\\x-4=0\\x=4[/tex]
x values are 4 and 7
Now we check the values using original equation and verify it
[tex]x=4\\\sqrt{x-3} +5=x\\\sqrt{4-3} +5=4\\\\6=4 \\[/tex]
x=4 does not satisfies the original equation
Now we check x=7
[tex]\sqrt{7-3} +5=7\\7=7[/tex]
x=7 is the solution
Learn more : brainly.com/question/13401711
1. Define Line, Line segment and Ray.
2. Define Complementary Angles.
3. Define Supplementary Angles.
4. Define Adjacent Angles.
5. Define Linear Pair
Answer:
A line is a straight path of points that has no beginning or end
A line segment is a portion of a line that has two endpoints
A ray is a portion of a line which has one endpoint and extends forever in one direction
Complementary angles are two angles whose sum adds up to 90 degrees
Supplementary angles are two angles that add up to 360 degrees
John sold a car for 60,000 dollars and made a profit of 20%.what is the cost price of the car
Answer:
72,000
Step-by-step explanation:
20% of 60,000 = 12,000
60,000 + 12,000 = 72,000
Chow,...!
Which pair of transformations moves quadrilateral 1 to quadrilateral 2?
a:Reflect over the line y = -3, then rotate it 90 degrees counterclockwise about the origin
b:Reflect it over the x-axis, then rotate it 180 degrees about the origin
c:Rotate it 90 degrees counterclockwise about the point (-3, -3) then translate it 8 units to the right
d:Translate it 8 units to the right, then reflect it over the line y=-3
Answer:
Answer choices:
a. Reflection takes it to Q2, further rotation takes it back to Q3
Incorrectb. Reflection takes it to Q2, rotation takes to Q4 with right shape.
Correctc. Rotation takes is to Q4, translation keeps in same Q but with wrong shape
Incorrectd. Translation takes it to Q4, reflection keeps it in same Q but with wrong shape
IncorrectBecause the shape is present in Q3
So reflection over x axis takes it to Q2
Then 180° rotation about origin takes it to Q4
Hence Option B is correct
for f(x)=x^2-2, find the rate of change on the interval [-2,4]
Answer:
2
Step-by-step explanation:
steps are in picture.Ignore hand writing
Can someone pls explain how to do order of operations
Answer:
do you mean PEMDAS?
Step-by-step explanation:
parenthesis, exponents, multiplication, division, addition, and subtraction
you solve in order and go left to right
if you give an equation i can explain it better, but it's basically that
Step-by-step explanation:
1- solve operation in parantheses or brackets.
2- we solve any exponent
3- solve all multiplication and division left to right
Write an expression that represents the perimeter of the rectangle
below. Write your answer in STANDARD FORM
5x + 2
2.x
[tex]\\ \sf\longmapsto Area=Length(Breadth)
[tex]\\ \sf\longmapsto Area=(5x+2)2x[/tex]
[tex]\\ \sf\longmapsto Area=10x^2+4x[/tex]
simplify expression -3√2+ 3√8
Answer:
3√2
Step-by-step explanation:
−3√2+3√8
=3√2 (Decimal: 4.242641)
Answer:
3√2
~~~~~~~~~~~~~~~
-3√2+ 3√8
-3√2+ 6√2
3√2
Step-by-step explanation:
how do I solve -2a = -24
Answer:
12
Step-by-step explanation:
you would divide -2 on both sides to isolate the variable a leaving you with 12
Answer:
12
Step-by-step explanation:
-2a= -24
or, a= -24/-2
:. a= 12
if LN=9x-39, LM=47, and MN=3x+10 find MN
Answer:
11
Step-by-step explanation:
ML+LN=MN solve for x
(47)+(9x-39)=(3x+10) = 1/3
x = 1/3
plug into MN
3(1/3)+10=11
8 - 4s = s + 13
what is s?
Answer:
-5s = 5
S= - 1
Step-by-step explanation:
Udndkdjdndjxbdnekdjdndkdjdjd
What's the area of a room that 18ft by 10ft
Answer:
180 square feet
Step-by-step explanation:
Square footage is calculated by multiplying width by length. So if a room is 18 foot wide by 10 foot long, 18 x 10 = 180 square feet.
Select all numbers that will round to 600 when rounded to the nearest hundred.
550
575
596
602
667
Answer:
550, 575, 596, 602
Answer:
Anything greater or equal to 550 will be rounded to 600
and anything less than 650 will be rounded to 600,
So the numbers are:-
550
575
596
602
triangle ABC is congruent to triangle DBC
Answer:
hi i am new in this app can u help
Step-by-step explanation:
hi miss can you help
