The algebra of the four basic operations

MATHEMATICS

Grade 9

ALGEBRA AND GEOMETRY

Module 6

THE ALGEBRA OF THE FOUR BASIC OPERATIONS

Activity 1

To refresh understanding of conventions in algebra as applied to addition and subtraction

[LO 1.2, 1.6]

A We will first have a look at terms.

Remember, terms are separated by + or –. In each of the following, say how many terms there are:

1. a + 5

2. 2a2

3. 5a (a+1)

4. $\frac{3a-1}{a^2}+\mathrm{5a}$

In the next exercise you must collect like terms to simplify the expression:

1. 5a + 2a

2. 2a2 + 3a – a2

3. 3x – 6 + x + 11

4. 2a(a–1) – 2a2

B Adding expressions

Example:

Add 3x + 4 by x + 5.

(x + 5) + (3x + 4) Write, with brackets, as sum.

x + 5 + 3x + 4 Remove brackets, with care.

4x + 9 Collect like terms.

In this exercise, add the two given expressions:

1. 7a + 3 and a + 2

2. 5x – 2 and 6 – 3x

3. x + ½ and 4x – 3½

4. a2 + 2a + 6 and a – 3 + a2

5. 4a2 – a – 3 and 1 + 3a – 5a2

C Subtracting expressions

Study the following examples very carefully:

Subtract 3x – 5 from 7x + 2.

(7x + 2) – (3x – 5)

Notice that 3x – 5 comes second, after the minus.

7x + 2 – 3x + 5

The minus in front of the bracket makes a difference!

4x + 7

Collecting like terms.

Calculate 5a – 1 minus 7a + 12: (5a – 1) – (7a + 12)

5a – 1 – 7a – 12

–2a – 13

D Mixed problems

Do the following exercise (remember to simplify your answer as far as possible):

1. Add 2a – 1 to 5a + 2.

2. Find the sum of 6x + 5 and 2 – 3x.

3. What is 3a – 2a2 plus a2 – 6a?

4. (x2 + x) + (x + x2) = . . .

5. Calculate (3a – 5) – (a – 2).

6. Subtract 12a + 2 from 1 + 7a.

7. How much is 4x2 + 4x less than 6x2 – 13x?

8. How much is 4x2 + 4x more than 6x2 – 13x?

9. What is the difference between 8x + 3 and 2x +1?

Use appropriate techniques to simplify the following expressions:

1. x2 + 5x2 – 3x + 7x – 2 + 8

2. 7a2 – 12a + 2a2 – 5 + a – 3

3. (a2 – 4) + (5a + 3) + (7a2 + 4a)

4. (2x – x2) – (4x2 – 12) – (3x – 5)

5. (x2 + 5x2 – 3x) + (7x – 2 + 8)

6. 7a2 – (12a + 2a2 – 5) + a – 3

7. (a2 – 4) + 5a + 3 + (7a2 + 4a)

8. (2x – x2) – 4x2 – 12 – (3x – 5)

9. x2 + 5x2 – 3x + (7x – 2 + 8)

10. 7a2 – 12a + 2a2 – (5 + a – 3)

11. a2 – 4 + 5a + 3 + 7a2 + 4a

12. (2x – x2) – [(4x2 – 12) – (3x – 5)]

Here are the answers for the last 12 problems:

1. 6x2 + 4x + 6

2. 9a2 – 11a – 8

3. 8a2 + 9a – 1

4. – 5x2 – x + 17

5. 6x2 + 4x + 6

6. 5a2 – 11a + 2

7. 8a2 + 9a – 1

8. – 5x2 – x – 7

9. 6x2 + 4x + 6

10. 9a2 – 13a – 2

11. 8a2 + 9a – 1

12. – 5x2 + 5x + 7

Activity 2

To multiply certain polynomials by using brackets and the distributive principle

[LO 1.2, 1.6, 2.7]

A monomial has one term; a binomial has two terms; a trinomial has three terms.

A Multiplying monomials.

Brackets are often used.

Examples:

2a × 5a = 10a2

3a3 × 2a × 4a2 = 24 a6

4ab × 9a2 × (–2a) × b = –36a4b2

a × 2a × 4 × (3a2)3 = a × 2a × 4 × 3a2 × 3a2 × 3a2 = 126a8

(2ab2)3 × (a2bc)2 × (2bc)2 = (2ab2) (2ab2) (2ab2) × (a2bc) (a2bc) × (2bc) (2bc) = 32a7b10c4

Always check that your answer is in the simplest form.

Exercise:

1. (3x) (5x2)

(x3) (–2x)

(2x)2 (4)

(ax)2 (bx2) (cx2)2

B Monomial × binomial

Brackets are essential.

Examples:

5(2a + 1) means multiply 5 by 2a as well as by 1. 5 (2a + 1) = 10a + 5

Make sure that you work correctly with your signs.