IGCSE Additional Math 3.1. Use remainder and factor theorems Topic Practice

CAIE IGCSE Additional Math 3.1. Use remainder and factor theorems question practice helps you revise this syllabus point with the course map in view. Use this page to focus on one topic, check the style of questions available, and connect each attempt back to the knowledge area it is testing.

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3.1.1—Use remainder and factor theorems

[Maximum number: 5]

The polynomial p is such that $\mathrm{p}(x)=x^{3}+a x^{2}+b x-2, \quad$ where a and b are constants.
It is given that:
- x+2 is a factor of p(x)
- when p(x) is divided by x-3 the remainder is 40.

Find the values of a and b.

[Maximum number: 5]

It is given that $\mathrm{p}(x)=a x^{3}-7 x^{2}-b x+9$ , where a and b are constants. x-3 is a factor of p(x).
When p(x) is divided by x+2 the remainder is -35.
Find the values of a and b.

[Maximum number: 6]

The polynomial $\mathrm{p}(x)=6 x^{3}+a x^{2}+b x+2$ , where a and b are integers, has a factor of x-2.

(a)

Given that p(1)=-2 p(0), find the values of a and b.

[ 4 ]

(b)

Using your values of a and b,

[ 2 ]

(i)

find the remainder when p(x) is divided by 2 x-1

[ 2 ]

[Maximum number: 3]

DO NOT USE A CALCULATOR IN THIS QUESTION.

The polynomial p is such that $\mathrm{p}(x)=6 x^{3}-35 x^{2}+34 x+45$ .

Find p(x) in the form (2 x-5) q(x)+r, where q(x) is a polynomial and r is a constant.

[Maximum number: 6]

The polynomial $p(x)=m x^{3}-29 x^{2}+39 x+n$ , where m and n are constants, has a factor 3 x-1, and remainder 6 when divided by x-1. Show that x-2 is a factor of p(x).

[Maximum number: 3]

DO NOT USE A CALCULATOR IN THIS QUESTION.

The polynomial p is defined by $\mathrm{p}(x)=a x^{3}-3 x^{2}-3 x+b$ , where a and b are constants.

Given that x=2 and x=-1 are roots of the equation p(x)=0, find a and b.

[Maximum number: 3]

The polynomial p is given by $\mathrm{p}(x)=a^{2} x^{3}+2 a x^{2}+a x+2$ , where a is a positive integer. It is given that 2 x+1 is a factor of p(x).

Find the value of a.

[Maximum number: 6]

The polynomial p is such that $\mathrm{p}(x)=a x^{3}+11 x^{2}+b x+c$ , where a, b and c are integers. It is given that $\mathrm{p}^{\prime}(0)=12$ .
It is also given that x+3 is a factor of p.
When p is divided by x-1 the remainder is 16.
Find the values of a, b and c.

[Maximum number: 6]

The polynomial p is such that $\mathrm{p}(x)=a x^{3}+b x^{2}-19 x+c$ , where a, b and c are integers. It is given that x+2 is a factor of p(x). When p(x) is divided by x+1 the remainder is 20.

(a)

Show that 7 a-3 b=39.

It is also given that when $\mathrm{p}^{\prime}(x)$ is divided by x-1 the remainder is 1.

[ 3 ]

(b)

Find the values of a, b and c.

[ 3 ]

[Maximum number: 4]

The polynomial p is such that $\quad \mathrm{p}(x)=3 x^{3}-7 x^{2}+a x+b$ , where a and b are integers.
It is given that $\mathrm{p}^{\prime}(-1)=21$ and that x-2 is a factor of p(x).

Find the values of a and b.

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