MULTIPLICATION OF ALGEBRAIC EXPRESSIONS MODULE – 4 MULTIPLICATION OF ALGEBRAIC EXPRESSIONS

Multiplying a binomial by a trinomial. ( LO 2 AS 4) Factorizing trinomials. (LO 2 AS 4) Factorizing by grouping in pairs. (LO 2 AS 4) Simplifying algebraic fractions with monomial denominators. (LO 2 AS 4)

The Distributive law (Revision of grade 9) x (y + z) = xy + xz To verify this law, we can substitute given numbers for the variables. Suppose that x = 2, y = 3 and z = 4. Then the left side will have the following value: x(y + z) = 2(3 + 4) = 2(7) = 14 The right side will have the same value as the left side: xy + xz = 2 x 3 + 2 x 4 = 6 + 8 = 14 Therefore 2(3 + 4) = 2 x 3 + 2 x 4 The product of two binomials (Revision of grade 9) Consider the product (a + b) (c + d). We can apply the distributive law as follows: (a + b)(c + d) = (a + b) c + (a + b) d = c (a + b) + d (a + b) = ca + cb + da + db = ac + ad + bc + bd This can also be done using the FOIL method. Here you must first multiply the first terms in each bracket. Then you multiply the outer terms, then the inner terms and finally the last terms.

Example 1 Expand and simplify the following: (a) (x + 2) (x + 3)

EXERCISE 3 1. Solve for x and y by using the method of substitution: (a) x - y = 2 and 2x + y = 10 (b) y - 3x = - 2 and 7x - 2y = 8 (c) 3x + 5y = 8 and x - 2y = - l (d) 7x - 3y = 41 and 3x – y = 17

2. Solve for x and y by using the method of elimination: (a) x – y = 2 and 2x + y = l0 (b) x + y = -5 and 3x + y = -9 (c) x + 2y = 5 and x – y = - l (d) 3x + 5y = 8 and x - 2y = -l (e) 2x - 3y = l0 and 4x + 5y = 42

3. Solve for x and y using any method of your choice: (a) x + y = l and x - 2y = l (b) 3x + 2y = 2 and 5x - 2y = - l8 (c) x + 4y = 14 and 3x + 2y = 12 (d) 2y - 3x = 7 and 4y - 5x = 21 (e) 3x + 2y = 6 and 5x + 3y = 11

D. QUADRATIC EQUATIONS A quadratic equation has the form ax + bx + c = 0 and has at most two real solutions. It is important to factories the quadratic expression and then apply what is called the zero factor law, which states that if a.b = 0, then either a = 0 or b = 0. This is due to the fact that if you multiply any number by zero the answer will always be zero.

For example, the zero factor law can be applied to the equation x(x —2) = 0 as follows: x – 2 = 0 (Zero factor law) x = 2

2. Expand and simplify: (a) (x + 3)(x - 3) (b) (x - 6)(x + 6) (c) (2x - l)(2x + l) (d) (4x + 9)(4x - 9) (e) (3x - 2y)(3x + 2y) (f) (4a b + 3)(4a b - 3) (g) (6 – 3x y)(6 + 3x y) (h) (2x – 3 + y)(2x – 3 – y) (i) (a - 4)(a + 4) (j) (1 – a )(1 – a )(1 + a)

3. Expand and simplify: (a) 2x(3x - 4y) - (7x - 2xy) (b) (5y + 1) - (3y + 4)(2 - 3y) (c) (2x + y) - (3x - 2y) + (x - 4y)(x + 4y) (d) (8m - 3n)(4m + n) - (n - 3m)(n + 3m) (e) x + (x - 3y)(x + 3y) (f) (3a + b)(3a - b)(2a + 5b)

The product of a binomial and a trinomial Example 4 Expand and simplify the following: (a) Method 1: Pattern: (term one)(bracket with trinomial) – (term two)(bracket with trinomial)

Reminder, be very careful with your signs.

(b)

Now, want to try? Let’s go!

EXERCISE 2 Expand and simplify: (a) (x + 1)(x + 2x + 3) (b) (x - 1)(x - 2x + 3) (c) (2x + 4)(x - 3x + 1) (d) (2x - 4)(x - 3x + 1) (e) (3x—y)(2x + 4xy – y ) (f) (a + 2b)(4a - 3ab + b ) (g) (3x - 2y)(9x + 6xy + 4y ) (h) (3x + 2y)(9x - 6xy + 4y ) (i) (5m - 2n)(2m + 4mn - 7n ) (j) (2a + 3b ) (k) (2a - 3b )

FACTORISATION OF ALGEBRAIC EXPRESSIONS The golden rule of factorization is to always look for the HCF first Taking out the Highest Common Factor (Revision of grade 9) Factorization is the reverse of multiplication. Consider the reverse procedure of the distributive law: ab + ac = a(b + c) The highest common factor (a) has been “taken out” of the expression ab + ac and the expression is said to be factorized as the product a (b + c).

Example 5 Factorise the following expressions: (a) 12x + 8x The factors of 12 are: 1; 2; 3; 4; 6; and 12 The factors of 8 are: 1; 2; 4; and 8 Therefore the highest common factor between 12 and 8 is 4. The expression x can be written as: Therefore the highest number of common x’s between the two expressions is . In other words, the highest common factor between and is Therefore, the highest common factor between and is . We can now take out the highest common factor (HCF) as follows and thus factorize the expression by writing it as a product of terms: 12x + 8x = 4x . 3 + 4x . 2x = 4x (3 + 2x ) (b) 15a b - 3ab The HCF= 3ab 15a b - 3ab = 3ab x5a b - 3ab x1 = 3ab (5a b - 1)

(c) 2a(x + y) - 3b(x + y) The HCF = (x + y) 2a (x + y) -3b(x + y) = (x + y)(2a – 3b)

2. Factorize: (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n)

Difference of two squares (Revision of grade 9) Consider the product : The reverse process: is called the factorisation of the difference of two squares. Another way of seeing this type of factorisation is: where a > 0, b > 0

Example 7 REMEMBER TO ALWAYS LOK FOR THE HCF FIRST Factorise fully:

(c) (d)

EXERCISE 4 Factorise fully: REMEMBER TO FIRST LOOK FOR THE HCF (a) (b) (d) (e) (f)

(g) (h) (i) (j) (k) (l) (m) (n) (o) (p) (q) (r)

Quadratic Trinomials Consider the product: (m + a) (m + b) By multiplying out, it is clear that this product will become: (m + a)(m + b)

So the expression can be factorized as: (m + a)(m + b) For example, the trinomial can be factorized as follows: Write the last term, 8, as the product of two numbers .

The options are: 1 x 8 4 x 2 The middle term (a + b) is now obtained by adding the numbers of one of the above options. The obvious choice will be the option 4x2 because the sum of the numbers 4 and 2 gives 6. Therefore:

So the trick to factorizing trinomials is as follows: Write down the last term as the product of two numbers Find the two numbers (using the appropriate numbers from one of the products) which gets the middle term by adding or subtracting. Check that when you multiply these numbers you get the last term.

Example 8 Factorize: (a) The last term can be written as the following products: 1 x 60, 30 x 2, 15 x 4, 10 x 6, 12 x 5, 20 x 3 We now need to get - 7 from one of the options above. Using 12 x 5 will enable us to get - 7 since - 12 + 5 = - 7 which is the middle term and (- 12)(5) = - 60 which is the last term. Therefore:

(b) The last term can be written as the following products: 3 x 2, 1 x 6 We now need to get the middle term - 5 from one of the options above. Try the option 3 x 2. Clearly – 3 - 2 = - 5 which is the middle term and - 3 x - 2 = + 6 which is the last term Notice that the option 1 x 6 will not work because even though + 1 - 6 = - 5 is the middle term, 1 x - 6 = - 6, is not the last term. Therefore:

Notice: If the sign of the last term of a trinomial is positive, the signs in the brackets are the same (see example 8 a). If the sign of the last term of a trinomial is negative, the signs in the brackets are different, i.e. both positive or both negative (c) Here it is necessary to first take out the highest common factor:

The last term of the trinomial in the brackets can be written as the following products: 1 x 8, 4 x 2 The option 1 x 8 will work because: + 1 - 8 = - 7 which is the middle term, and ( + l) (- 8) = - 8 which is the last term. Therefore:

Notice that the signs in the brackets are different (because the sign of the last term, - 8, is negative). In summary then, apply the following procedure when factorizing trinomials: Take out the highest common factor if necessary. Write down the last term as the product of two numbers. Find the two numbers (using the appropriate numbers from one of the products) which gets the middle term by adding or subtracting. Check that when you multiply these numbers you get the last term. If the sign of the last term of a trinomial is positive, the signs in the brackets are the same. If the sign of the last term of a trinomial is negative, the signs in the brackets are different (both positive and both negative).

(d) The last term can be written as the following products: 1 x 12, 4 x 3, 6 x 2 The signs in the brackets must be different because the sign of the last term of the trinomial is negative. The option 4 x 3 will work because: + 4 - 3 = +1 which is the middle term, and (+ 4)(- 3) = - 12 which is the last term Therefore:

(e) The last term can be written as the following products: 1 x 24, 12 x 2, 8 x 3, 6 x 4 The signs in the brackets must be the same because the sign of the last term, + 24, is positive. The option 6 x 4 will work because: - 6 - 4 = - 10, which is the middle term, and (- 6) (- 4) = + 24, which is the last term. Notice that the option 12 x 2 will not work because even though - 12 + 2 = - 10 is the middle term, -12 x 2 = - 24, is not the last term. Therefore:

EXERCISE 5 Factorize fully : ( REMEMBER TO LOOK FOR THE HCF FIRST) (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (o)

2. Factorize fully: (REMEMBER TO LOOK FOR THE HCF FIRST) (a) (b) (c) (d) (e)

3. Explain why the trinomial cannot factorize. 4. Explain why the following factorizations are incorrect: (a) (b)

More advanced quadratic trinomials Example 9 Consider the trinomial . The method to factorize this trinomial is a little more involved than with the previous trinomials. A suggested method is as follows.

Step 1 CHECK FOR THE HCF

Step 2 Write down the brackets and the factors of the first term and the factors of the last term. (7p 1)(3p 4)

Step 3 Now multiply the innermost and the outermost terms.

Step 4 To find the middle term do the following: - 3p + 28p = + 25p

Step 5 Complete the factors (7p – 1)(3p + 4) Note: This method involves trial and error and you need to keep trying different options until you get ones that will work.

Example 10 Factorize :

Step 1: Find the factors of the first term of each bracket. (4y 1)(3y 2)

Step 2 Multiply the innermost and the outermost terms. 3y 8y

Step 3 Now add them to find the middle term- 3y - 8y = - 11y

Step 4 Complete the factorization process (4y - 1)(3y - 2)

Example 11 Factorize

Step 1: Find the factors of the first and last terms of each bracket (12a b)(2a b)

Step 2: Multiply the innermost and the outermost terms. 2ab 12ab

Step 3: Add these products to find the middle term - 12ab + 2ab = - 10ab

Step4: Complete the factorization process (12a + b)(2a – b)

EXERCISE 6 1. Factorize fully: REMEMBER TO LOOK FOR THE HCF FIRST (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l)

2. Factorize fully: (a) (b) (c) (d) (e) (f)

Factorization by grouping in pairs Example 12 Factorize: Method 1 Group the first two terms together and the last two together. Factorize each pair separately and then take out the common bracket:

Method 2 Group the first and third terms together and the second and fourth terms together. Factorize each pair separately and then take out the common bracket:

Before discussing the next examples, we need to deal with the concept of “taking out a negative”. Consider the following expressions: - (x - y) - (x + y) and = - x + y = - x - y It is clear from the above that: - x + y = - (x - y) and – x – y = - (x + y)

Therefore, whenever you “take out a negative sign” when factorizing an expression, the middle sign will always change in the brackets.

Example 13 Factorize the following expressions fully: (a) (b) (c)

Exercise 7 1. Factorize the following: (a) (b) (c) (d) (e)

2. (a) Prove that (b) Hence, prove by means of factorization, that 3. Consider the expression:

Show how the method of factorizing the expression is different from the method of expanding and simplifying.

SIMPLIFICATION OF ALGEBRAIC FRACTIONS Multiplication and division of algebraic fractions Example 14

Simplify the following expressions: (a) (b)

(c) Whenever the numerator contains two or more terms, factorize the expression in the numerator and simplify:

EXERCISE 8 Simplify the following: (a) (b) (c) (d) (e) (f) (g) (h) (i) (j)

Addition and subtraction of algebraic fractions Whenever you add or subtract algebraic fractions, the first thing you need to do is to determine the lowest common denominator (LCD). You then have to convert each fraction to an equivalent fraction so that each fraction’s denominator is the same as the LCD, The last thing to do is to write the numerators over the LCD using the rule

Example 15 Simplify: The multiples of 4 are: 4, 8, 12, 16, 20…. The lowest common multiple of these three numbers is 12. With the variables and the term with the highest exponent will be used in z the LCD. In this case, the term used will be. Since there are no other terms in you will just use z in the LCD. The LCD will therefore be: The trick is to now change each fraction so that the denominators of each fraction will be equal to the LCD.

EXERCISE 9 1. Simplify, the following: (a) (b) (c) (d) (e) (f)

ASSESSMENT TASKS Expand and simplify: Factorize: JOURNAL ENTRY Your friend, who missed the classes on factorization due to going overseas, wants you to explain the difference between expanding and simplifying expressions and the concept of factorizing expressions. Use the following example to show the difference. Expand and simplify: Factorize:

SPOT THE ERROR

In this question, attempts have been made to answer the given questions. Serious errors have been made. State in words what errors have been made and then redo the question correctly.

Attempt at the question Errors made Correct answer 1. Expand and simplify: 2. Factorize: 3.Expand and simplify: 4. Factorize:

5.Simplify: 6.Factorise: 7.Expand and simplify 8. Expand simplify:

Assessment rubric Completely correct response 2 Still developing 1 Incorrect