2023HSCShort answer

Q11 (c)

3 marks

Question

Consider the polynomial

P(x)=x^3+ax^2+bx-12,

where $a$ and $b$ are real numbers.

It is given that $x+1$ is a factor of $P(x)$ and that, when $P(x)$ is divided by $x-2$ , the remainder is $-18$ .

Find $a$ and $b$ .

Try the question before revealing the answer.

Solution

Criteria	Marks
Provides correct solution	3
Obtains two equations for $a$ and $b$ , or equivalent merit	2
Observes that $P(-1)=0$ or $P(2)=-18$ , or equivalent merit	1

Sample answer:

$P(-1)=0$ by factor theorem

\begin{aligned} (-1)^3+a(-1)^2+b(-1)-12&=0\\ -1+a-b-12&=0\\ a-b&=13 \qquad (1) \end{aligned}

$P(2)=-18$ by remainder theorem

\begin{aligned} 2^3+a(2)^2+b(2)-12&=-18\\ 8+4a+2b-12&=-18\\ 4a+2b&=-14\\ 2a+b&=-7 \qquad (2) \end{aligned}

\begin{aligned} (1)+(2):\quad 3a&=6\\ a&=2 \end{aligned}

Using this in (1):

\begin{aligned} 2-b&=13\\ b&=-11 \end{aligned}

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Syllabus alignment

Functions

PolynomialsUse polynomial division, the factor and remainder theorems, zeros and graphs to solve polynomial problems.

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