Algebra, Functions & Graphs

13 methods · 48 questions

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When asked to simplify a sum, difference or quotient of algebraic fractions, never expand the numerator. Factorise every quadratic into its linear brackets first, then cancel common factors top and bottom before combining terms over a common denominator.

Trigger: a sum, difference or quotient of rational expressions to be simplified.

Instances:
(i) factorise every quadratic expression into linear brackets;
(ii) cancel common linear factors top and bottom to reduce the degree;
(iii) dividing by a fraction is multiplying by its reciprocal, which brings more linear brackets into contact for cancellation.

Linked questions (3)

NSAA 2019 Section 1, Part A (Mathematics questions) · Q6

Which of the following is a simplification of

4 - \frac{x ( 3 x + 1 )}{x ^{2} ( 3 x ^{2} - 2 x - 1 )}

A $\frac{12 x ^{3} - 8 x ^{2} - 7 x - 1}{x ( 3 x - 1 ) ( x - 1 )}$
B $\frac{4 x ^{2} + 4 x - 1}{x ( x + 1 )}$
C $\frac{4 x ^{2} + 4 x + 1}{x ( x + 1 )}$
D $\frac{4 x ^{2} - 4 x - 1}{x ( x - 1 )}$
E $\frac{4 x ^{2} - 4 x + 1}{x ( x - 1 )}$
F $\frac{12 x ^{3} - 8 x ^{2} - x + 1}{x ( 3 x - 1 ) ( x - 1 )}$

Worked solution

Answer:D

Factorise the quadratic in the denominator. Two numbers with product

3 \times (- 1) = - 3

and sum

- 2

are

- 3

and

1

, so

3 x^{2} - 2 x - 1 = 3 x^{2} - 3 x + x - 1 = 3 x (x - 1) + (x - 1) = (3 x + 1) (x - 1) .

The fraction therefore simplifies, cancelling one factor of

x

and the common factor

3 x + 1

\frac{x ( 3 x + 1 )}{x ^{2} ( 3 x + 1 ) ( x - 1 )} = \frac{1}{x ( x - 1 )} .

Finally, put the whole expression over the common denominator

x (x - 1)

4 - \frac{1}{x ( x - 1 )} = \frac{4 x ( x - 1 ) - 1}{x ( x - 1 )} = \frac{4 x ^{2} - 4 x - 1}{x ( x - 1 )} .

The answer is D.

NSAA 2017 Section 1, Part A (Mathematics questions) · Q7

Which one of the following is a simplification of

2 - \frac{x ^{2} ( 9 x ^{2} - 4 )}{x ^{3} ( 2 - 3 x )}

A $- 1 - \frac{2}{x}$
B $- 1 + \frac{2}{x}$
C $5 - \frac{2}{x}$
D $5 + \frac{2}{x}$
E $5 - \frac{3}{x}$
F $5 + \frac{3}{x}$

Worked solution

Answer:D

Factorise the bracket in the numerator as a difference of two squares:

9 x^{2} - 4 = (3 x - 2) (3 x + 2) .

The bracket in the denominator is the same factor reversed,

2 - 3 x = - (3 x - 2)

, so those factors cancel and the powers of

x

reduce from

x^{2}

over

x^{3}

\frac{1}{x}

\frac{x ^{2} ( 3 x - 2 ) ( 3 x + 2 )}{x ^{3} \cdot ( - ( 3 x - 2 ) )} = - \frac{3 x + 2}{x} .

Subtracting a negative quantity adds it back:

2 - (- \frac{3 x + 2}{x}) = 2 + \frac{3 x + 2}{x} = 2 + 3 + \frac{2}{x} = 5 + \frac{2}{x} .

Testing

x = 1

in the original expression gives

2 - \frac{1 \times 5}{1 \times ( - 1 )} = 7

, and

5 + \frac{2}{1} = 7

.

The answer is D.

NSAA 2016 Section 1, Part A (Mathematics questions) · Q11

Which one of the following is a simplification of

4 + \frac{4 - x ^{2}}{x ^{2} - 2 x}

A $3 - \frac{2}{x}$
B $3 + \frac{2}{x}$
C $4 - \frac{2}{x}$
D $4 + \frac{2}{x}$
E $5 - \frac{2}{x}$
F $5 + \frac{2}{x}$

Worked solution

Answer:A

Factorise the numerator and the denominator of the fraction:

4 - x^{2} = (2 - x) (2 + x) = - (x - 2) (x + 2), x^{2} - 2 x = x (x - 2) .

For

x \neq = 0

and

x \neq = 2

the common factor

x - 2

cancels:

\frac{4 - x ^{2}}{x ^{2} - 2 x} = \frac{- ( x - 2 ) ( x + 2 )}{x ( x - 2 )} = - \frac{x + 2}{x} = - 1 - \frac{2}{x} .

Adding the leading

4

gives

4 - 1 - \frac{2}{x} = 3 - \frac{2}{x} .

The cancellation is where the constant term drops from

4

3

, which is exactly what the distractors miss.

As a check, at

x = 1

the original expression is

4 + \frac{3}{- 1} = 1

, and

3 - \frac{2}{1} = 1

.

The answer is A.