Divide the following polynomials using a synthetic division \$(- x^3 + x^2 + 11x +10) / (x + 2)\$.

The given polynomials

\$(- x^3 + x^2 + 11x +10) / (x + 2)\$

Write down the coefficients of the polynomial:

−1, 1, 11, 10

Since we’re dividing by x + 2, we put −2 outside the division box.

Now, let’s perform the synthetic division:
1. Bring down the leading coefficient, which is

-1

2.Multiply the divisor, -2, by the result from step 1, which gives us

2

3.Add the coefficient from the polynomial in the first column to the result from step 2. So,

1 + 2 = 3.

4.Multiply the divisor, -2, by the result from step 3, which gives us

-6

5.Add the coefficient from the polynomial in the second column to the result from step 4. So,

11 + (- 6) = 5

6.Multiply the divisor, -2, by the result from step 5, which gives us

-10

7.Add the coefficient from the polynomial in the third column to the result from step 6. So,

10 + (-10) = 0

So, the quotient is \$−x^2 + 3x + 5\$ and the remainder is 0.

Therefore, \$(−x^3 + x^2 + 11x + 10)\$ divided by (x + 2) equals \$−x^2 + 3x + 5\$.

Because the quotient given, \$x^2 − 3x−5\$, does not match the correct quotient obtained through synthetic division

Quotient \$x^2 - 3x - 5\$ and the remainder is 0

Option B is not correct because it provides a different quotient, \$x^2 + 3x + 5\$

Quotient \$x^2 + 3x+ 5\$ and the remainder is 0

Wrong: Because the sign of the constant term in the quotient is incorrect (− 5 instead of + 5)

Quotient \$- x^2 + 3x - 5\$ and the remainder is 0

Option D aligns with the quotient from synthetic division. It matches the polynomial \$-x^2 + 3x + 5\$

Quotient \$- x^2 + 3x+ 5\$ and the remainder is 0

Option

D

Can you summarize what you’ve understood in the above steps?