How do you collect and distribute this?

Find a polynomial $f(x)$ of degree $5$ such that both of these properties hold: $\bullet$ $f(x)$ is divisible by $x^3$. $\bullet

frosja888 [35]

There seems to be one character missing. But I gather that f(x) needs to satisfy

• $x^3$ divides $f(x)$

• $(x-1)^3$ divides $f(x)^2$

I'll also assume f(x) is monic, meaning the coefficient of the leading term is 1, or

$f(x) = x^5 + \cdots$

Since $x^3$ divides $f(x)$ , and

$f(x) = x^3 p(x)$

where $p(x)$ is degree-2, and we can write it as

$f(x)=x^3 (x^2+ax+b)$

Now, we have

$f(x)^2 = \left(x^3p(x)\right)^2 = x^6 p(x)^2$

so if $(x - 1)^3$ divides $f(x)^2$ , then $p(x)$ is degree-2, so $p(x)^2$ is degree-4, and we can write

$p(x)^2 = (x-1)^3 q(x)$

where $q(x)$ is degree-1.

Expanding the left side gives

$p(x)^2 = x^4 + 2ax^3 + (a^2+2b)x^2 + 2abx + b^2$

and dividing by $(x-1)^3$ leaves no remainder. If we actually compute the quotient, we wind up with

$\dfrac{p(x)^2}{(x-1)^3} = \underbrace{x + 2a + 3}_{q(x)} + \dfrac{(a^2+6a+2b+6)x^2 + (2ab-6a-8)x +2a+b^2+3}{(x-1)^3}$

If the remainder is supposed to be zero, then

$\begin{cases}a^2+6a+2b+6 = 0 \\ 2ab-6a-8 = 0 \\ 2a+b^2+3 = 0\end{cases}$

Adding these equations together and grouping terms, we get

$(a^2+2ab+b^2) + (2a+2b) + (6-8+3) = 0 \\\\ (a+b)^2 + 2(a+b) + 1 = 0 \\\\ (a+b+1)^2 = 0 \implies a+b = -1$

Then $b=-1-a$ , and you can solve for a and b by substituting this into any of the three equations above. For instance,

$2a+(-1-a)^2 + 3 = 0 \\\\ a^2 + 4a + 4 = 0 \\\\ (a+2)^2 = 0 \implies a=-2 \implies b=1$

So, we end up with

$p(x) = x^2 - 2x + 1 \\\\ \implies f(x) = x^3 (x^2 - 2x + 1) = \boxed{x^5-2x^4+x^3}$

6 0

2 years ago

Marni and Sharika partnered together in a dance-a-thon to raise money for the Humane Society. Marni raised $18 plus $0.80 per mi

Snowcat [4.5K]

I'm assuming that the 3434 is a typo error. It should be 34.

Marni: 18 + 0.80m
Sharika: 12 + 0.9m

x = Sharika's minutes
x + 34 = Marni's minutes

[18 + 0.80(x+34)] + [12 + 0.90(x)] = 264
18 + 0.80x + 27.2 + 12 + 0.90x = 264
0.80x + 0.90x = 264 - 18 - 27.2 - 12
1.70x = 206.80
x = 206.80/1.70
x = 121.65 or 122

x + 34 = 122 + 34 = 156 minutes.

5 0

3 years ago

What is these answers

lianna [129]

1. 15
2. 9/14
3. 5/8
4. 2
5. 8.4*10 exponent -3
6. 90
7. 0.0025
8. -60
9. 15

For 9 I’m not that sure but I’m guessing it’s an absolute value question???

Btw can I have brainliest? Lol

7 0

2 years ago

Read 2 more answers

A triangle has two sides that measures 7 cm which of the following cannot be the measure of the third side

klasskru [66]

Any side that is greater than 7 + 7 = 14 will not make a triangle.

x > 14 will not create a triangle.

7 0

3 years ago

He two square roots of the number. 64

tankabanditka [31]

Answer:

The square route of 64 is 8.