Question

use synthetic division to find the result when $3x^3 + 14x^2 - 10x - 28$ is divided by $x + 5$. if there is a remainder, express the result in the form $q(x) + \frac{r(x)}{b(x)}$.

Explanation:

Step1: Identify the root

For the divisor \(x + 5\), the root \(c\) is \(-5\) (since \(x+5 = x - (-5)\)). The coefficients of the dividend \(3x^{3}+14x^{2}-10x - 28\) are \(3\), \(14\), \(-10\), \(-28\).

Step2: Set up synthetic division

Set up the synthetic division as follows:
Bring down the first coefficient \(3\).
Multiply \(3\) by \(-5\) to get \(-15\). Add this to the second coefficient \(14\): \(14+(-15)=-1\).
Multiply \(-1\) by \(-5\) to get \(5\). Add this to the third coefficient \(-10\): \(-10 + 5=-5\).
Multiply \(-5\) by \(-5\) to get \(25\). Add this to the fourth coefficient \(-28\): \(-28+25=-3\).

Step3: Write the quotient and remainder

The synthetic division gives us the coefficients of the quotient polynomial and the remainder. The quotient \(q(x)\) is a quadratic polynomial with coefficients \(3\), \(-1\), \(-5\), so \(q(x)=3x^{2}-x - 5\). The remainder \(r(x)=-3\) and the divisor \(b(x)=x + 5\). So the result is \(3x^{2}-x - 5+\frac{-3}{x + 5}\) or \(3x^{2}-x - 5-\frac{3}{x + 5}\).

Answer:

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