Classwork Series and Exercises {Mathematics- JSS2}: Expansion of Algebraic Expressions

Directed Algebraic Terms

Remember that in algebra, letters stand for numbers. The number can be whole or fractional, positive or negative.

1. Just as 5a is short for 5 x a, so -5a is short for (-5) x a.
2. Just as m is short for 1 x m, so –m is short for (-1) x m.
3. Algebraic terms and numbers can be multiplied together. For example,

4 X (-3x) = (+4) x (-3) X x

                = -(4 x 3) X x = -12 X x = -12x

(-2y) x (-8y) = (-2) X y X (-8) X y

                       = (-2) X (-8) X x X y

                       = +(2 X 8) X y²

                       = +16y² or just 16²

4. Division with directed numbers is also possible. For example,

18a ÷ (-6) = (+18) X a/(-6)

                   = – (18/6) X a

                   = (-3) X a = -3a

-33x² /-3x = (-33) X x X x/(-3) X x

                  = +(33/3) X x = 11x

Substitution

Since letters in algebra stand for numbers, it is always possible to substitute values for the letters. Just as in football, the manager might substitute player N0 4 for players No 23, so in algebraa we might substitute a value such as 8 or -5 for an unknown x.

Example

Find the value of

• 4x b. xy – 5y when x = 2 and y = 3.

Solution

• Substitute the value 2 for x, i.e. use the value 2 instead of x.

When x = 2, 4x = 4 X x

                             = 4 X 2 = 8

• xy – 5y = x X y – 5 X y

 When x = 2 and y = 3,

   xy – 5y = 2 X 3 – 5 X 3

                 = 6 – 15 = -9

Example

What is the value of p – q/p when p = -5 and q = +10?

Notice that p – q/p is the same as(p – q)/q.

Simplify the top line before dividing.

When p = -5 and q = +10,

P – q/p = (-5) – (+10)/(-5) = -15/-5 = +(15/5) = +3

Removing Brackets

3 X (7 + 5) means first add 7 and 5, then multiply the result by 3. Suppose a pencil costs 70 naira and a rubber costs 50 naira.

Cost of a pencil and a rubber

= 70 naira + 50 naira

= (70 + 50) naira = 120 naira

If three students each buy a pencil and a rubber, then,

Total cost = 3 X (70 + 50) naira

                    = 3 X 120 naira

                    = 360 naira

There is another way to find the total cost. Three pencils cost 3 X 70 naira. Three rubbers cost 3 X 50 naira. Altogether,

total cost = 3 X 70 naira + 3 X 50 naira

                   = 210 naira + 150 naira

                   = 360 naira

Thus, 3 X (70 + 50) = 3 X 70 + 3 X 50.

This shows that brackets can be removed by multiplying the three into both the 70 and the 50.

Usually, we do not write the multiplication sign. We just write 3(70 + 50).

Say 3(70 + 50) as ‘3 into (70 + 50)’

3(70 + 50) = 3 X 70 + 3 X 50

In general, using letters for numbers,

a(x + y) = ax + ay

Notice also that:

3(70 + 50) = 3 X 20 = 60

and 3 X 70 – 3 X 50 = 210 – 150 = 60

Thus, 3(70 – 50) = 3 X 70 – 3 X 50

Again, using letters for numbers,

 a(x + y) = ax + ay

Example

Remove brackets from the following:

1. 8(2c + 3d)
2. 4y(3x – 5)

Solution

• 8(2c + 3d) = 8 X 2c + 8 X 3d

                    = 16c + 24d       

• 4y(3x – 5) = 4y X 3x – 4y X 5

                        = 12xy – 20y

Expanding Algebraic Expression

The expression (a + 2)(b – 5) means (a + 2) X (b – 5) means (a + 2) X (b – 5). The terms in the first bracket, (a + 2), multiply each term in the second bracket, (b – 5). Just as:

X(b – 5) = bx – 5x

So, writing (a + 2) instead of x,

(a + 2)(b – 5) = b(a + 2) – 5(a + 2)

The brackets on the right-hand side can now be removed.

(a + 2)(b – 5) = b(a + 2) – 5(a + 2)

                         = ab + 2b – 5a – 10

ab + 2b – 5a – 10 is the product of (a + 2) X (b – 5). We often say that the expansion of (a + 2)(b – 5) is:

ab + 2b – 5a – 10

Example

Expand the following:

1. (a + b)(c + d)
2. (6 – x)(3 + y)

Solution

1. (a + b)(c + d) = c(a + b) + d(a + b)

                           = ac _ bc + ad + db

1. (6 – x)(3 + y) = 3(6 – x) + y(6 – x)

                            = 18 – 3x + 6y – xy

We sometimes call this binomial expansion, since each bracket contain two terms (bi-nomial means two-names).

EXERCISES

Lets see how much you’ve learnt, attach the following answers to the comment below

Expand the following:

1. (p + q)(r + s)
2. (x + 8)(y + 3)
3. (4 + 5a)(3b + a)
4. (a – b)(c + d)

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