The algebra of the four basic operations  (Page 2/5)

4a(2a + 1) = 8a + 4a

–5a(2a + 1) = –10a2 – 5a

a2(–3a2 – 2a) = –3a4 – 2a3

–7a(2a – 3) = –14a2 + 21a

Note: We have turned an expression in factors into an expression in terms. Another way of saying the same thing is: A product expression has been turned into a sum expression.

Exercise:

1. 3x (2x + 4)

x2 (5x – 2)

–4x (x2 – 3x)

(3a + 3a2) (3a)

C Monomial × trinomial

Examples:

5a(5 + 2a – a2) = 25a + 10a2 – 5a3

– ½ (10x5 + 2a4 – 8a3) = – 5x5 – a4 +4a3

Exercise:

3x (2x2 – x + 2)

–ab2 (–bc + 3abc – a2c)

12a ( ¼ + 2a + ½ a2)

Also try: 4. 4x (5 – 2x + 4x2 – 3x3 + x4)

D Binomial × binomial

Each term of the first binomial must be multiplied by each term of the second binomial.

(3x + 2) (5x + 4) = (3x)(5x) + (3x)(4) + (2)(5x) + (2)(4) = 15x2 + 12x + 10x + 8

= 15x2 + 22x + 8

Always check that your answer has been simplified.

Here is a cat–face picture to help you remember how to multiply two binomials:

The left ear says multiply the first term of the first binomial with the first term of the second binomial.

The chin says multiply the first term of the first binomial with the second term of the second binomial.

The mouth says multiply the second term of the first binomial with the first term of the second binomial.

The right ear says multiply the second term of the first binomial with the second term of the second binomial.

There are some very important patterns in the following exercise – think about them.

Exercise:

(a + b) (c + d)

(2a – 3b) (–c + 2d)

(a2 + 2a) (b 2 –3b)

(a + b) (a + b)

(x2 + 2x) (x2 + 2x)

(3x – 1) (3x – 1)

(a + b) (a – b)

(2y + 3) (2y – 3)

(2a2 + 3b) (2a2 – 3b)

(a + 2) (a + 3)

(5x2 + 2x) (x2 – x)

(–2a + 4b) (5a – 3b)

E Binomial × polynomial

Example:

(2a + 3) (a3 – 3a2 + 2a – 3) = 2a4 – 6a3 + 4a2 – 6a + 3a3 – 9a2 + 6a – 9

= 2a4 – 3a3 – 5a2 – 9 (simplified)

Exercise:

(x2 – 3x) (x2 + 5x – 3)

(b + 1) (3b2 – b + 11)

(a – 4) (5 + 2a – b + 2c)

(–a + 2) (a + b + c – 3d)

Activity 3

To find factors of certain algebraic expressions

[LO 1.6, 2.1, 2.7]

A Understanding what factors are

This table shows the factors of some monomials

You can write the factors in any order, but if you stick to the usual order your work will be easier. Two lists of factors in the table are not in the usual order – rewrite them in order.

B Finding common factors of binomials

Take the binomial 6ab + 3ac.

The factors of 6ab are 2 × 3 × a × b and the factors of 3ac are 3 × a × c.

The factors that appear in both 6ab and 3ac are 3 and a – they are called common factors.

We can now use brackets to group the factors into the part that is common and the rest, as follows:

6ab = 3a × 2b and 3ac = 3a × c

Now we can factorise 6ab + 3ac. This is how to set it out:

6ab + 3ac = 3a (2b + c).

An expression in terms has been written as an expression in factors.

Or: A sum expression has been turned into a product expression.

Here are some more examples: