Answer:

Step-by-step explanation:

We have studied procedures for working with fractions in earlier grades.

ab×cd=acbd(b≠0;d≠0)

ab+cb=a+cb(b≠0)

ab÷cd=ab×dc=adbc(b≠0;c≠0;d≠0)

Note: dividing by a fraction is the same as MULTIPLYING by the reciprocal of the fraction.

In some cases of simplifying an algebraic EXPRESSION, the expression will be a fraction. For EXAMPLE,

x2+3xx+3

has a quadratic binomial in the numerator and a LINEAR binomial in the denominator. We have to apply the different factorisation methods in order to factorise the numerator and the denominator before we can simplify the expression.

x2+3xx+3=x(x+3)x+3=x(x≠−3)

If x=−3 then the denominator, x+3=0 and the fraction is undefined.                                       WORKED EXAMPLE 18: SIMPLIFYING FRACTIONS

Simplify:

ax−b+x−abax2−abx,(x≠0;x≠b)

Use grouping to factorise the numerator and take out the common factor ax in the denominator

(ax−ab)+(x−b)ax2−abx=a(x−b)+(x−b)ax(x−b)

Take out common factor (x−b) in the numerator

=(x−b)(a+1)ax(x−b)

Cancel the common factor in the numerator and the denominator to give the final answer

=a+1ax

WORKED EXAMPLE 19: SIMPLIFYING FRACTIONS

Simplify:

x2−x−2x2−4÷x2+xx2+2x,(x≠0;x≠±2)

Factorise the numerator and denominator

=(x+1)(x−2)(x+2)(x−2)÷x(x+1)x(x+2)

Change the division sign and multiply by the reciprocal

=(x+1)(x−2)(x+2)(x−2)×x(x+2)x(x+1)

Write the final answer

=1

WORKED EXAMPLE 20: SIMPLIFYING FRACTIONS

Simplify:

x−2x2−4+x2x−2−x3+x−4x2−4,(x≠±2)

Factorise the denominators

x−2(x+2)(x−2)+x2x−2−x3+x−4(x+2)(x−2)

Make all denominators the same so that we can add or subtract the fractions

The lowest common denominator is (x−2)(x+2).

x−2(x+2)(x−2)+(x2)(x+2)(x+2)(x−2)−x3+x−4(x+2)(x−2)

Write as one fraction

x−2+(x2)(x+2)−(x3+x−4)(x+2)(x−2)

Simplify

x−2+x3+2x2−x3−x+4(x+2)(x−2)=2x2+2(x+2)(x−2)

Take out the common factor and write the final answer

2(x2+1)(x+2)(x−2)

WORKED EXAMPLE 21: SIMPLIFYING FRACTIONS

Simplify:

2x2−x+x2+x+1x3−1−xx2−1,(x≠0;x≠±1)

Factorise the numerator and denominator

2x(x−1)+(x2+x+1)(x−1)(x2+x+1)−x(x−1)(x+1)

Simplify and find the common denominator

2(x+1)+x(x+1)−x2x(x−1)(x+1)

Write the final answer

2x+2+x2+x−x2x(x−1)(x+1)=3x+2x(x−1)(x+1)