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

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

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Answer 1 · 2018-07-11T09:57:28

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

Explanation:

Let's perform the synthetic division

$a a a a$ $color(white)(aaaa)$ $- 3$ $-3$ $∣$ $|$ $a a a a$ $color(white)(aaaa)$ $1$ $1$ $a a a a$ $color(white)(aaaa)$ $19$ $19$ $a a a a a a$ $color(white)(aaaaaa)$ $108$ $108$ $a a a a$ $color(white)(aaaa)$ $236$ $236$ $a a a a a$ $color(white)(aaaaa)$ $176$ $176$

$a a a a a a a$ $color(white)(aaaaaaa)$ $∣$ $|$ $a a a a$ $color(white)(aaaa)$ $a a a a$ $color(white)(aaaa)$ $- 3$ $-3$ $a a a a a$ $color(white)(aaaaa)$ $- 48$ $-48$ $a a a$ $color(white)(aaa)$ $- 180$ $-180$ $a a a$ $color(white)(aaa)$ $- 168$ $-168$

$a a a a a a a a a$ $color(white)(aaaaaaaaa)$ _________________________________________________________##

$a a a a a a a$ $color(white)(aaaaaaa)$ $∣$ $|$ $a a a a$ $color(white)(aaaa)$ $1$ $1$ $a a a a$ $color(white)(aaaa)$ $16$ $16$ $a a a a a a$ $color(white)(aaaaaa)$ $60$ $60$ $a a a a a a$ $color(white)(aaaaaa)$ $56$ $56$ $a a a a a a$ $color(white)(aaaaaa)$ $8$ $color(red)(8)$

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

Apply the remainder theorem for verification

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

$f (x) = (x - c) q (x) + r$ $f(x)=(x-c)q(x)+r$

Let $x = c$ $x=c$

Then,

$f (c) = 0 + r$ $f(c)=0+r$

Here,

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

Therefore,

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

$= 8$ $=8$

The remainder is $= 8$ $=8$

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

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How do you synthetic division and the Remainder Theorem to find P(–3) if P(x)=x4+19x3+108x2+236x+176P(x) = x^4 + 19x^3 + 108x^2 + 236x + 176?

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Explanation:

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