CAIE A-Level Mathematics A2 3.1 Algebra Questions

Practise Pure 3 algebra skills with modulus graphs, polynomial division, theorem use, partial fractions and binomial expansions.

Exam points

solve modulus inequalities using intersections, sign cases or squared forms
divide polynomials to find quotients, remainders and values of constants
expand binomial expressions and state the valid range using strict inequalities

Covered syllabus points

3.1.1—Modulus equations 3.1.2—Polynomial division 3.1.3—Factor and remainder theorems 3.1.4—Partial fractions 3.1.5—Binomial expansion

[Maximum number: 4]

Solve the inequality |2 x+3|>3|x+2|.

Square both sides:
$(2x+3)^2>9(x+2)^2.$
Then
$4x^2+12x+9>9x^2+36x+36,$
so
$5x^2+24x+27<0.$
Factorising,
(5x+9)(x+3)<0.
The critical values are x=-3 and $x=-\frac95$ . The product is negative between them, hence
$-3<x<-\frac95.$

B1 for the non-modular squared inequality or equivalent linear equations. M1 for solving the quadratic/equations. A1 for critical values -3 and $-\frac95$ . A1 for $-3<x<-\frac95$ , with strict inequalities.

Figure for Answer to Question 1 — CAIE A-Level Mathematics A2

[Maximum number: 3]

Find the quotient and remainder when $8 x^{3}+4 x^{2}+2 x+7$ is divided by $4 x^{2}+1$ .

Commence division and reach quotient of the form $2 x \pm 1$
Or by inspection $8 x^{3}+4 x^{2}+2 x+7=\left(4 x^{2}+1\right)(2 x \pm 1) +\mathrm{r}$
Or by inspection $8 x^{3}+4 x^{2}+2 x+7=\left(4 x^{2}+1\right)(2 x \pm 1)$
+r

Obtain (quotient) 2 x+1
Obtain (remainder) 6
3

[Maximum number: 2]

The polynomial $x^{3}+5 x^{2}+31 x+75$ is denoted by p(x).

Show that (x+3) is a factor of p(x).

Substitute x=-3 to obtain value of p(-3)
Obtain p(-3)=0 and hence given result
Alternative method for Question 10(a)

Divide p(x) by ( x+3 ) to obtain quotient $x^{2} \pm 2 x+\ldots$
Obtain quotient $x^{2}+2 x+25$ , with zero remainder and hence given result

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Covered syllabus points

Question 1

Question 2

Question 3