Question

I WILL GIVE BRAINLIEST IF YOU CAN ANSWER THESE TWO QUESTIONS!!!!

2. Darwin believes that x + b is a factor of (x + b)3 + (x + c)2 – (b - c), while Eunju
believes it is not. Who is correct? Justify your response.

3. How can you prove that x2 + 5x + 6 is a factor of x* + 5x3 + 2x2 – 20x – 24, without
using long or synthetic division?

Answer 1

Answer:

2. Eunju is right

3. $x^2+5x+6$ is a factor of $x^4+5x^3+2x^2-20x-24$.

Step-by-step explanation:

Polynomial Remainder Theorem

The polynomial remainder theorem states that the remainder of the division of a polynomial f(x) by (x-a) is equal to f(a).

As a consequence, if a polynomial is divisible by x-a, f(a)=0.

Part 1:

Let's make:

$f(x)=(x+b)^3+(x+c)^2-(b-c)$

To find out if x+b is a factor of f(x), we find f(-b):

$f(-b)=(-b+b)^3+(-b+c)^2-(b-c)$

Operating:

$f(-b)=(-b+c)^2-(b-c)$

The value of f(-b) is not zero. This means Eunju is right, x+b is not a factor of f(x).

Part 2:

We must find out if $x^2+5x+6$ is a factor of $x^4+5x^3+2x^2-20x-24$ without using long division or synthetic division.

We can use the polynomial remainder theorem again, but since the factor is not in the form (x-a), we can factor it as follows:

$x^2+5x+6 =(x+2)(x+3)$

Now we just apply the theorem twice. If both remainders are zero, then the assumption is true.

Let's make:

$f(x)=x^4+5x^3+2x^2-20x-24$

Find f(-2):

$f(-2)=(-2)^4+5(-2)^3+2(-2)^2-20(-2)-24$

$f(-2)=16-5*8+2*4+40-24 =16-40+8+40-24=0$

Find f(-3):

$f(-3)=(-3)^4+5(-3)^3+2(-3)^2-20(-3)-24$

$f(-3)=81-5*27+2*9+60-24$

$f(-3)=81-135+18+60-24 =0$

Since both f(-2) and f(-3) are zero, $x^2+5x+6$ is a factor of $x^4+5x^3+2x^2-20x-24$.