Question

The polynomial P(x) = x⁴ + ax³ - 7x² - 4ax +b has a factor (x+3) and remainder 60 when divided by (x -3). Find the values of a and b.

Answer 1

Hi,

a=0, b=42

Explanation:

1) we divide P(x) by (x-3)

`\begin {array}c&x^4&x^3&x^2&x&1\\---&--&--&--&--&--\\&1&a&-7&-4a&b\\x=3&&3&3a+9&9a+6&15a+18\\---&--&--&--&--&--\\&1&a+3&3a+2&5a+6&15a+18+b\\\end {array}`

`P(x)=q(x)*(x-3)+60\\\\15a+b=42\\q(x)=x^3+(a+3)x^2+(3a+2)x+(5a+6)\\`

2) (x+3) is a factor of q(x)

We divide q(x) by (x+3) and remainder =0

`\begin{array}&c&c&c&c&x^3&x^2&x&1\\---&--&--&--&--\\&1&a+3&3a+2&5a+6\\x=-3&&-3&-3a&-6\\---&--&--&--&--\\&1&a&2&5a\\\end {array}\\5a=0\Longrightarrow\ a=0\\15a+b=42\Longrightarrow\ b=42\\`

Proof:

`p(x)=x^4-7x^2+42=(x-3)*(x^3+3x^+2x+6)+60\\\\x^3+3x^+2x+6=(x+3)(x^2+2)\\\\\boxed{p(x)=(x-3)(x+3)(x^2+2)+60}`