Find the sum of (3a -4c),(-5a+2b)(=11a+6c) , identify type of expression
Answers
Answer:
1 The algebra of the four basic operations
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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. 3a−1a2+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.
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)
REFER TO ATTACHMENT
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