CAIE A-Level Mathematics 2.1.3 Factor and Remainder Theorems

Practise evaluating p(a) for factors and remainders, forming simultaneous equations for unknown coefficients and dividing by known linear factors to complete polynomial solutions.

Exam points

substitute the zero of ax + b into p(x), equating the result to zero for a factor
translate each stated remainder into p(a) = r and solve the resulting coefficient equations
divide by a known factor and factorise the quotient, then classify any remaining real roots

[Maximum number: 2]

Factorise $x^{4}-32 x+48$ .

Include $(x-2)^{2}$ as a factor
M1
Must be a product of factors only
SC B1 for $\left(x^{2}-4 x+4\right)\left(x^{2}+4 x+12\right)$
Conclude $(x-2)^{2}\left(x^{2}+4 x+12\right)$
A1
isw any attempt to factorise the quotient.

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2.1.3—Factor and remainder theorems question 1