Exercise: Expand and simplify

2 x−

√

1 x

3

Think Quick: . Factorize 10x2 y − 6xy 2 .
See Example 1.16 on page 17.

Factorizing Quadratics Key: x2 + (α + β)x + αβ = (x + α)(x + β) Basic Factorization Factorization is all about finding factors.
Turn back to the bottom of page 16.

Note. You will have learned a technique for applying this result at school. Revise it!
See also Example 1.17 on page 17.

Example Factorize 6x2 + 17x + 12.

Standard Factorizations The following are handy to remember.

• a3 ± b3 = (a ± b)(a2 ∓ ab + b2 ) • an − bn = (a − b)(an−1 + an−2 b + . . . + abn−2 + bn−1 )

Exercise: Factorize the following 1. −5m3 − 10m2

2. 16x4 − y 4

Algebraic Fractions Exercise: 3. a3 + a2 − a − 1 Recall:

• To simplify a fraction, factorize the numerator and denominator, then cancel out any common factors.

• To add/subtract fractions, find a common denominator. Express all frac4. 5x2 − 7x − 6 Note. tions with that common denominator, then add/subtract the numerators.

c a+c a + = b d b+d

Examples 1.

x x+y 3x + 6 6x + 12

Exercise: Simplify the expression

2.

3 x 2 − + . x − 2 x x2 − 3x + 2

3.

2 1 − x − 1 x2 − 1

You should do Exercises 1.3.2, Questions 11-15.

1.3.3 Equations, inequalities and identities Solving Equations
Turn to page 20.

Exercise: Solve the equation

7 6 = . 1 − 2x x−2

Solving linear equations is a skill you should have already mastered.

Think Quick: Solve

1 = 1 for t. t−5

Solving equations that arise from worded problems is a skill you should continue to practice.
See Example 1.19 on page 20.

Solving Quadratic Equations A quadratic equation has the form

Example

2x + 3x2 − 7 = 0

ax2 + bx + c = 0 where a, b, c ∈ R, a = 0.

There are two main methods for solving quadratic equations:

• formula • factorisation
The Quadratic Formula
See equation (1.8)