Edexcel A-Level Mathematics AS P2.2 Algebra and Functions Questions

Practise using polynomial division, remainders and factors to find constants, quotient forms and fully factorised expressions.

Exam points

apply the remainder theorem to build equations for unknown coefficients in f(x)
divide by a linear factor to find Q(x) and the constant remainder

Covered syllabus points

P2.2.1 - Simple algebraic division

[Maximum number: 2]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

\mathrm{f}(z)=4 z^{3}+p z^{2}-24 z+108

where p is a constant.
Given that -3 is a root of the equation f(z)=0

determine the value of p

$\mathrm{f}(-3)=0 \Rightarrow 4(-3)^3+p(-3)^2-24(-3)+108=0 \Rightarrow p=\ldots$ p=-8
(2)

[Maximum number: 2]

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

\mathrm{f}(z)=z^{3}-13 z^{2}+59 z+p \quad p \in \mathbb{Z}

Given that z=3 is a root of the equation f(z)=0

show that p=-87

$f(3)=3^3-13(3)^2+59(3)+p=87+p$ 87+p=0p=-87

[Maximum number: 2]

A curve C has equation y=f(x) where

f(x)=(2-k x)^{5}

and k is a constant.
Given that when f(x) is divided by (4 x-5) the remainder is $\frac{243}{32}$

show that $k=\frac{2}{5}$

$f\left(\frac54\right)=\left(2-k\cdot\frac54\right)^5=\frac{243}{32}$ $2-\frac{5k}{4}=\sqrt[5]{\frac{243}{32}}=\frac32$ $\frac{5k}{4}=\frac12,\qquad k=\frac25.$
M1 A1*

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Covered syllabus points

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