# How do you synthetic division and the Remainder Theorem to find P(–3) if P(x) = x^4 + 19x^3 + 108x^2 + 236x + 176?

Jul 11, 2018

The remainder is $8$ and the quotient is $= {x}^{3} + 16 {x}^{2} + 60 x + 56$

#### Explanation:

Let's perform the synthetic division

$\textcolor{w h i t e}{a a a a}$$- 3$$|$$\textcolor{w h i t e}{a a a a}$$1$$\textcolor{w h i t e}{a a a a}$$19$$\textcolor{w h i t e}{a a a a a a}$$108$$\textcolor{w h i t e}{a a a a}$$236$$\textcolor{w h i t e}{a a a a a}$$176$

$\textcolor{w h i t e}{a a a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a}$$- 3$$\textcolor{w h i t e}{a a a a a}$$- 48$$\textcolor{w h i t e}{a a a}$$- 180$$\textcolor{w h i t e}{a a a}$$- 168$

$\textcolor{w h i t e}{a a a a a a a a a}$_________________________________________________________

$\textcolor{w h i t e}{a a a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$1$$\textcolor{w h i t e}{a a a a}$$16$$\textcolor{w h i t e}{a a a a a a}$$60$$\textcolor{w h i t e}{a a a a a a}$$56$$\textcolor{w h i t e}{a a a a a a}$$\textcolor{red}{8}$

The remainder is $8$ and the quotient is $= {x}^{3} + 16 {x}^{2} + 60 x + 56$

Apply the remainder theorem for verification

When a polynomial $f \left(x\right)$ is divided by $\left(x - c\right)$, we get

$f \left(x\right) = \left(x - c\right) q \left(x\right) + r$

Let $x = c$

Then,

$f \left(c\right) = 0 + r$

Here,

$f \left(x\right) = {x}^{4} + 19 {x}^{3} + 108 {x}^{2} + 236 x + 176$

Therefore,

$f \left(- 3\right) = {\left(- 3\right)}^{4} + 19 \cdot {\left(- 3\right)}^{3} + 108 {\left(- 3\right)}^{2} + 236 \cdot \left(- 3\right) + 176$

$= 8$

The remainder is $= 8$