Question

G(x) = 2x^3 - px^2 + x - 5p

The remainder when g(x) is divided by (x-2) is four times larger than the remainder when g(x) is divided by (x + 1). Use this information to solve for the value of p

Answer 1

p = -2

Explanation:

Given polynomial:

`g(x) = 2x^3 - px^2 + x - 5p`

The remainder theorem states that when a polynomial p(x) is divided by a linear polynomial (x - c), then the remainder is equal to p(c).

Therefore, if the polynomial g(x) is divided by (x - 2), then the remainder is equal to g(2).

`\begin{aligned}\textsf{Remainder\;1:}\quad g(2)&=2(2)^3-p(2)^2+2-5p\\&=2(8)-p(4)+2-5p\\&=16-4p+2-5p\\&=18-9p\end{aligned}`

If the polynomial g(x) is divided by (x + 1), then the remainder is equal to g(-1).

`\begin{aligned}\textsf{Remainder\;2:}\quad g(-1)&=2(-1)^3-p(-1)^2+(-1)-5p\\&=2(-1)-p(1)-1-5p\\&=-2-p-1-5p\\&=-3-6p\end{aligned}`

Given remainder 1 is four times larger than remainder 2:

`18-9p=4(-3-6p)`

Solve for p:

`\begin{aligned} 18-9p&=4(-3-6p)\\18-9p&=-12-24p\\15p&=-30\\p&=-2\\\end{aligned}`

Therefore, the value of p is -2.