Question

question
use synthetic division to find the result when ( x^4 + 8x^3 + 24x^2 + 25x - 14 ) is divided by ( x + 3 ). if there is a remainder, express the result in the form ( q(x) + \frac{r(x)}{b(x)} ).

Explanation:

Step1: Identify root for divisor

For \(x + 3\), the root is \(x=-3\) (since \(x + 3=0\Rightarrow x=-3\)).

Step2: Set up synthetic division

Write coefficients of dividend \(x^{4}+8x^{3}+24x^{2}+25x - 14\): \(1, 8, 24, 25, -14\).
Bring down the first coefficient: \(1\).

Step3: Multiply and add (1st iteration)

Multiply \(1\times(-3)=-3\). Add to next coefficient: \(8+(-3)=5\).

Step4: Multiply and add (2nd iteration)

Multiply \(5\times(-3)=-15\). Add to next coefficient: \(24+(-15)=9\).

Step5: Multiply and add (3rd iteration)

Multiply \(9\times(-3)=-27\). Add to next coefficient: \(25+(-27)=-2\).

Step6: Multiply and add (4th iteration)

Multiply \(-2\times(-3)=6\). Add to last coefficient: \(-14 + 6=-8\) (this is the remainder \(r(x)=-8\)).

Step7: Form quotient polynomial

The coefficients of quotient \(q(x)\) are \(1, 5, 9, -2\), so \(q(x)=x^{3}+5x^{2}+9x - 2\), divisor \(b(x)=x + 3\), remainder \(r(x)=-8\).

Step8: Write in required form

\(q(x)+\frac{r(x)}{b(x)}=x^{3}+5x^{2}+9x - 2+\frac{-8}{x + 3}=x^{3}+5x^{2}+9x - 2-\frac{8}{x + 3}\)

Answer:

\(x^{3}+5x^{2}+9x - 2-\frac{8}{x + 3}\)