ALGEBRAIC EXPRESSIONS - Products

Mathematical expressions are just like sentences and their parts have special names. You should be familiar with the following words used to describe the parts of mathematical expressions.

3x + 7xy − 53 = 0

Name           Examples

term             3x ; 7xy ; − 53

expression    3x2 + 7xy − 53

coefficient     3; 7

exponent       2; 1; 3

base              x; y; 5

constant        3; 7; 5

variable         x; y

equation       3x + 7xy − 53 = 0

Multiplying a monomial and a binomial 

A monomial is an expression with one term, for example, 3x or y. A binomial is an expression with two terms, for example, ax + b or cx + d.

Example : Simplifying brackets

QUESTION Simplify: 2(a − 1) − 3(a − 1).

SOLUTION

2(a − 1) − 3(a − 1) = 2(a) + 2(−1) + (−3)(a) + (−3)(−1)

= 2a − 2 − 3a + 3

= −a − 1

Multiplying two binomials

Here we multiply (or expand) two linear binomials:

(ax + b)(cx + d)

(ax + b)(cx + d) = (ax)(cx) + (ax)d + b(cx) + bd

                         = acx2 + adx + bcx + bd

                         = acx2 + x(ad + bc) + bd

Example 8: Multiplying two binomials
QUESTION
Find the product: (3x − 2)(5x + 8).
SOLUTION
(3x − 2)(5x + 8) = (3x)(5x) + (3x)(8) + (−2)(5x) + (−2)(8)
                           = 15x2 + 24x − 10x − 16
                           = 15x2 + 14x − 16
The product of two identical binomials is known as the square of the binomial and is
written as:
(ax + b)2 = a2x2 + 2abx + b2
If the two terms are of the form ax + b, and ax − b then their product is:
(ax + b)(ax − b) = a2x2 − b2
This product yields the difference of two squares.


Multiplying a binomial and a trinomial
A trinomial is an expression with three terms, for example, ax2 + bx + c. Now we learn how to multiply a binomial and a trinomial.
To find the product of a binomial and a trinomial, multiply out the brackets:

(A + B)(C + D + E) = A(C + D + E) + B(C + D + E)

Example 9: Multiplying a binomial and a trinomial
QUESTION
Find the product: (x − 1)(x2 − 2x + 1).
SOLUTION
Step 1 : Expand the bracket
(x − 1)(x2 − 2x + 1) = x(x2 − 2x + 1) − 1(x2 − 2x + 1)
                                 = x3 − 2x2 + x − x2 + 2x − 1
Step 2 : Simplify
                                 = x3 − 3x2 + 3x − 1

Exercise
Expand the following products:
1. 2y(y + 4)
2. (y + 5)(y + 2)
3. (2 − t)(1 − 2t)
4. (x − 4)(x + 4)
5. (2p + 9)(3p + 1)
6. (3k − 2)(k + 6)
7. (s + 6)2
8. −(7 − x)(7 + x)
9. (3x − 1)(3x + 1)
10. (7k + 2)(3 − 2k)
11. (1 − 4x)2
12. (−3 − y)(5 − y)
13. (8 − x)(8 + x)
14. (9 + x)2
15. (−2y2 − 4y + 11)(5y − 12)