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 β 2 and 6 β 3x
3. x + Β½ and 4x β 3Β½
4. a2 + 2a + 6 and a β 3 + a2
5. 4a2 β a β 3 and 1 + 3a β 5a2
C Subtracting expressions
Study the following examples very carefully:
Subtract 3x β 5 from 7x + 2.
(7x + 2) β (3x β 5)
Notice that 3x β 5 comes second, after the minus.
7x + 2 β 3x + 5
The minus in front of the bracket makes a difference!
4x + 7
Collecting like terms.
Calculate 5a β 1 minus 7a + 12: (5a β 1) β (7a + 12)
5a β 1 β 7a β 12
β2a β 13
D Mixed problems
Do the following exercise (remember to simplify your answer as far as possible):
1. Add 2a β 1 to 5a + 2.
2. Find the sum of 6x + 5 and 2 β 3x.
3. What is 3a β 2a2 plus a2 β 6a?
4. (x2 + x) + (x + x2) = . . .
5. Calculate (3a β 5) β (a β 2).
6. Subtract 12a + 2 from 1 + 7a.
7. How much is 4x2 + 4x less than 6x2 β 13x?
8. How much is 4x2 + 4x more than 6x2 β 13x?
9. What is the difference between 8x + 3 and 2x +1?
Use appropriate techniques to simplify the following expressions:
1. x2 + 5x2 β 3x + 7x β 2 + 8
2. 7a2 β 12a + 2a2 β 5 + a β 3
3. (a2 β 4) + (5a + 3) + (7a2 + 4a)
4. (2x β x2) β (4x2 β 12) β (3x β 5)
5. (x2 + 5x2 β 3x) + (7x β 2 + 8)
6. 7a2 β (12a + 2a2 β 5) + a β 3
7. (a2 β 4) + 5a + 3 + (7a2 + 4a)
8. (2x β x2) β 4x2 β 12 β (3x β 5)
9. x2 + 5x2 β 3x + (7x β 2 + 8)
10. 7a2 β 12a + 2a2 β (5 + a β 3)
11. a2 β 4 + 5a + 3 + 7a2 + 4a
12. (2x β x2) β [(4x2 β 12) β (3x β 5)]
Here are the answers for the last 12 problems:
1. 6x2 + 4x + 6
2. 9a2 β 11a β 8
3. 8a2 + 9a β 1
4. β 5x2 β x + 17
5. 6x2 + 4x + 6
6. 5a2 β 11a + 2
7. 8a2 + 9a β 1
8. β 5x2 β x β 7
9. 6x2 + 4x + 6
10. 9a2 β 13a β 2
11. 8a2 + 9a β 1
12. β 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 Γ 5a = 10a2
3a3 Γ 2a Γ 4a2 = 24 a6
4ab Γ 9a2 Γ (β2a) Γ b = β36a4b2
a Γ 2a Γ 4 Γ (3a2)3 = a Γ 2a Γ 4 Γ 3a2 Γ 3a2 Γ 3a2 = 126a8
(2ab2)3 Γ (a2bc)2 Γ (2bc)2 = (2ab2) (2ab2) (2ab2) Γ (a2bc) (a2bc) Γ (2bc) (2bc) = 32a7b10c4
Always check that your answer is in the simplest form.