CAIE A-Level Mathematics AS 2.1 Algebra Questions

Practise solving modulus equations and inequalities and applying polynomial division, factor and remainder theorems to determine coefficients, factors and real roots.

Exam points

split a modulus relation at its critical points and combine only intervals satisfying all conditions
divide a polynomial by a linear or quadratic divisor and retain the stated remainder
apply p(a) = 0 or p(a) = r to form coefficient equations and complete the factorisation

Covered syllabus points

2.1.1—Modulus equations 2.1.2—Polynomial division 2.1.3—Factor and remainder theorems

[Maximum number: 4]

Solve the inequality |5 x+7|>|2 x-3|.

Solve 5 x+7=2 x-3 to obtain $-\frac{10}{3}$
Or inequality.

Attempt solution of linear equation where 5 x and 2 x have
different signs
Or inequality.

Obtain $-\frac{4}{7}$
State $x<-\frac{10}{3}, x>-\frac{4}{7}$
A0 if ' ... and . . . ' used.

Alternative Method for Question 1

State or imply non-modulus equation $(5 x+7)^{2}=(2 x-3)^{2}$
Or inequality.

Attempt solution of three-term quadratic equation
Or inequality.

Obtain $-\frac{10}{3}$ and $-\frac{4}{7}$
State $x<-\frac{10}{3}, x>-\frac{4}{7}$
A0 if ' ... and . . .' used.

[Maximum number: 3]

The polynomial p(x) is defined by $\mathrm{p}(x)=9 x^{3}+18 x^{2}+5 x+4$ .

Find the quotient when p(x) is divided by (3 x+2), and show that the remainder is 6 .

Carry out division at least as far as $3 x^{2}+k_{1} x$
Or equivalent (inspection, ...).

Obtain quotient $3 x^{2}+4 x-1$
Confirm remainder is 6
Answer given - necessary detail needed.

SC B1 for use of the factor theorem to show remainder is 6 if no other marks are awarded.

Alternative Method for Question 5(a)

Synthetic division
-2 / 3 & 9 & 18 & 5 & 4

& & -6 & 8 & -2

& 9 & 12 & -3 & 6
Obtain quotient $3 x^{2}+4 x-1$
Confirm remainder is 6

[Maximum number: 9]

The polynomial p(x) is defined by

\mathrm{p}(x)=6 x^{3}+a x^{2}+b x-20

where a and b are constants. It is given that (x+2) is a factor of p(x) and that the remainder is -11 when p(x) is divided by (x+1).

(a)

Find the values of a and b.

[ 5 ]

(b)

Hence factorise p(x), and determine the exact roots of the equation p(3 x)=0.

[ 4 ]

Divide by x+2 at least as far as the x term
or equivalent (inspection, ...).

Obtain $(x+2)^{2}(6 x-5)$
OE

Replace (or imply replacement of) x by 3 x in factorised form
Obtain $-\frac{2}{3}$ and $\frac{5}{18}$
and no others.

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Covered syllabus points

Question 1

Question 2

Question 3

Question (a)

Question (b)