CAIE IGCSE Additional Math 3.1 remainder and factor theorems

Use this theorem page to substitute roots or divisor values into p(x), form equations for constants and prove when a factor exists.

Exam points

Use p(a)=0 when x-a is a factor and p(a)=r when a remainder is given in context.
Form simultaneous equations for unknown polynomial constants such as a, b, m or n.
Show that a linear expression is a factor by proving the polynomial value is zero.

[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.

$a^{2}\left(-\frac{1}{2}\right)^{3}+2 a\left(-\frac{1}{2}\right)^{2}+a\left(-\frac{1}{2}\right)+2=0$ oe, soi

M1

Simplifies and solves for a

M1

FT their quadratic in a providing $\mathrm{p}\left(-\frac{1}{2}\right)=0$
oe attempted

a=4

A1

All question bank results loaded

3.1. Use remainder and factor theorems question 1