2. Multiplication of Algebraic Expressions
Einstein's famous equation involves multiplying algebraic terms
When we multiply algebraic expressions, we need to remember the Index Laws from the Numbers chapter.
Let's see how algebra multiplication works with a series of examples.
Example 1
Multiply x3(x4 + 5a)
Answer
We expand this out using the distributive law:
x3(x4 + 5a)
= x7 + 5ax3
We cannot simplify this answer any further. We present our answer in alphabetical order since it makes it easier to read when the problems become more involved.
Example 2
Multiply (x + 5)(a − 6)
Answer
We multiply this out as follows. We take each term of the first bracket and multiply them by the second bracket. Then we expand out the result.
(x + 5)(a − 6)
= x(a − 6) + 5(a − 6)
= ax − 6x + 5a − 30
We cannot do any more with this answer. There are no like terms, so we cannot simplify it in any way.
Example 3
Multiply (2x + 3)(x2 − x − 5)
Answer
We take the 2 terms of the first bracket and multiply both of them by the second bracket.
(2x + 3)(x2− x − 5)
= (2x)(x2 − x − 5) + (3)(x2 − x − 5)
= (2x3 − 2x2 − 10x) + (3x2 − 3x − 15)
= 2x3 + x2 − 13x − 15
This time we could collect together some like terms. There was:
−2x2 + 3x2 = x2
and
−10x − 3x = −13x
Example 4
Multiply `(x - 3)^2`
Answer
(x − 3)2
= (x − 3)(x − 3)
= x(x − 3) − 3(x − 3)
= x2 − 3x − 3x + 9
= x2 − 6x + 9
Important Note:
`(x - 3)^2`
is NOT equal to
`x^2 - 9`
Please take note of this! Many students confuse this idea and it's not surprising because sometimes the way we write math is not consistent. (See also Towards more meaningful math notation for a discussion on this issue.)
Example 5
Simplify 5x[−4 + 10(x − y)] + 7x
Answer
5x[−4 + 10(x − y)] + 7x
= 5x[−4 + 10x − 10y] + 7x
= −20x + 50x2 − 50xy + 7x
= −13x + 50x2 − 50xy
Example 6
Simplify (p − 1)(c − 1) + 2
Answer
p(c − 1) − 1(c − 1) + 2
= pc − p − c + 1 + 2
= pc − p − c + 3