OpenStax-CNX module: m38375 1
Products and Factors - Grade 10
*
[CAPS]
Free High School Science Texts Project
Based on Products and Factors by
Rory Adams
Free High School Science Texts Project
Mark Horner
Heather Williams
This work is produced by OpenStax-CNX and licensed under the
Creative Commons Attribution License 3.0
1 Introduction
In this chapter you will learn how to work with algebraic expressions. You will recap some of the work
on factorisation and multiplying out expressions that you learnt in earlier grades. This work will then be
extended upon for Grade 10.
2 Recap of Earlier Work
The following should be familiar. Examples are given as reminders.
2.1 Parts of an Expression
Mathematical expressions are just like sentences and their parts have special names. You should be familiar
with the following names used to describe the parts of a mathematical expression.
a · x k + b · x + cm = 0
(1)
d · yp + e · y + f ≤ 0
* Version 1.3: Jun 12, 2011 6:24 pm -0500
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http://cnx.org/content/m38375/1.3/
,OpenStax-CNX module: m38375 2
Name Examples (separated by commas)
term a · xk ,b · x, cm , d · y p , e · y , f
expression a · x k + b · x + cm , d · y p + e · y + f
coe
cient a, b, d, e
exponent (or index) k, p
base x, y , c
constant a, b, c, d, e, f
variable x, y
equation a · x k + b · x + cm = 0
inequality d · yp + e · y + f ≤ 0
binomial expression with two terms
trinomial expression with three terms
Table 1
2.2 Product of Two Binomials
A binomial is a mathematical expression with two terms, e.g. (ax + b) and (cx + d). If these two binomials
are multiplied, the following is the result:
(a · x + b) (c · x + d) = (ax) (c · x + d) + b (c · x + d)
= (ax) (cx) + (ax) d + b (cx) + b · d (2)
= ax2 + x (ad + bc) + bd
Exercise 1: Product of two binomials (Solution on p. 13.)
Find the product of (3x − 2) (5x + 8)
The product of two identical binomials is known as the square of the binomial and is written as:
2
(ax + b) = a2 x2 + 2abx + b2 (3)
If the two terms are ax + b and ax − b then their product is:
(ax + b) (ax − b) = a2 x2 − b2 (4)
This is known as the dierence of two squares.
2.3 Factorisation
Factorisation is the opposite of expanding brackets. For example expanding brackets would require 2 (x + 1)
to be written as 2x + 2. Factorisation would be to start with 2x + 2 and to end up with 2 (x + 1). In
previous grades, you factorised based on common factors and on dierence of squares.
http://cnx.org/content/m38375/1.3/
,OpenStax-CNX module: m38375 3
2.3.1 Common Factors
Factorising based on common factors relies on there being common factors between your terms. For example,
2x − 6x2 can be factorised as follows:
2x − 6x2 = 2x (1 − 3x) (5)
2.3.1.1 Investigation : Common Factors
Find the highest common factors of the following pairs of terms:
(a) 6y; 18x (b) 12mn; 8n (c) 3st; 4su (d) 18kl; 9kp (e) abc; ac
(f) 2xy; 4xyz (g) 3uv; 6u (h) 9xy; 15xz (i) 24xyz; 16yz (j) 3m; 45n
Table 2
2.3.2 Dierence of Two Squares
We have seen that:
(ax + b) (ax − b) = a2 x2 − b2 (6)
Since (6) is an equation, both sides are always equal. This means that an expression of the form:
a2 x2 − b2 (7)
can be factorised to
(ax + b) (ax − b) (8)
Therefore,
a2 x2 − b2 = (ax + b) (ax − b) (9)
For example, x2 −16 can be written as x2 − 42 which is a dierence of two squares. Therefore, the factors
of x2 − 16 are (x − 4) and (x + 4).
Exercise 2: Factorisation (Solution on p. 13.)
Factorise completely: b2 y 5 − 3aby 3
Exercise 3: Factorising binomials with a common bracket (Solution on p. 13.)
Factorise completely: 3a (a − 4) − 7 (a − 4)
Exercise 4: Factorising using a switch around in brackets (Solution on p. 13.)
Factorise 5 (a − 2) − b (2 − a)
http://cnx.org/content/m38375/1.3/
, OpenStax-CNX module: m38375 4
2.3.2.1 Recap
1. Find the products of:
(a) 2y (y + 4) (b) (y + 5) (y + 2) (c) (y + 2) (2y + 1)
(d) (y + 8) (y + 4) (e) (2y + 9) (3y + 1) (f) (3y − 2) (y + 6)
Table 3
Click here for the solution1
2. Factorise:
a. 2l + 2w
b. 12x + 32y
c. 6x2 + 2x + 10x3
d. 2xy 2 + xy 2 z + 3xy
e. −2ab2 − 4a2 b
Click here for the solution2
3. Factorise completely:
(a) 7a + 4 (b) 20a − 10 (c) 18ab − 3bc
(d) 12kj + 18kq (e) 16k2 − 4k (f) 3a2 + 6a − 18
(g) −6a − 24 (h) −2ab − 8a (i) 24kj − 16k2 j
(j) −a2 b − b2 a (k) 12k2 j + 24k2 j 2 (l) 72b2 q − 18b3 q 2
(m) 4 (y − 3) + k (3 − y) (n) a (a − 1) − 5 (a − 1) (o) bm (b + 4) − 6m (b + 4)
(p) a2 (a + 7) + a (a + 7) (q) 3b (b − 4) − 7 (4 − b) (r) a2 b2 c2 − 1
Table 4
Click here for the solution3
3 More Products
Khan Academy video on products of polynomials.
This media object is a Flash object. Please view or download it at
<http://www.youtube.com/v/fGThIRpWEE4&rel=0>
Figure 9
We have seen how to multiply two binomials in "Product of Two Binomials" (Section 2.2: Product of Two
Binomials). In this section, we learn how to multiply a binomial (expression with two terms) by a trinomial
1 http://www.fhsst.org/lxI
2 http://www.fhsst.org/lqV
3 http://www.fhsst.org/lqE
http://cnx.org/content/m38375/1.3/
Products and Factors - Grade 10
*
[CAPS]
Free High School Science Texts Project
Based on Products and Factors by
Rory Adams
Free High School Science Texts Project
Mark Horner
Heather Williams
This work is produced by OpenStax-CNX and licensed under the
Creative Commons Attribution License 3.0
1 Introduction
In this chapter you will learn how to work with algebraic expressions. You will recap some of the work
on factorisation and multiplying out expressions that you learnt in earlier grades. This work will then be
extended upon for Grade 10.
2 Recap of Earlier Work
The following should be familiar. Examples are given as reminders.
2.1 Parts of an Expression
Mathematical expressions are just like sentences and their parts have special names. You should be familiar
with the following names used to describe the parts of a mathematical expression.
a · x k + b · x + cm = 0
(1)
d · yp + e · y + f ≤ 0
* Version 1.3: Jun 12, 2011 6:24 pm -0500
http://cnx.org/content/m31483/1.3/
http://creativecommons.org/licenses/by/3.0/
http://cnx.org/content/m38375/1.3/
,OpenStax-CNX module: m38375 2
Name Examples (separated by commas)
term a · xk ,b · x, cm , d · y p , e · y , f
expression a · x k + b · x + cm , d · y p + e · y + f
coe
cient a, b, d, e
exponent (or index) k, p
base x, y , c
constant a, b, c, d, e, f
variable x, y
equation a · x k + b · x + cm = 0
inequality d · yp + e · y + f ≤ 0
binomial expression with two terms
trinomial expression with three terms
Table 1
2.2 Product of Two Binomials
A binomial is a mathematical expression with two terms, e.g. (ax + b) and (cx + d). If these two binomials
are multiplied, the following is the result:
(a · x + b) (c · x + d) = (ax) (c · x + d) + b (c · x + d)
= (ax) (cx) + (ax) d + b (cx) + b · d (2)
= ax2 + x (ad + bc) + bd
Exercise 1: Product of two binomials (Solution on p. 13.)
Find the product of (3x − 2) (5x + 8)
The product of two identical binomials is known as the square of the binomial and is written as:
2
(ax + b) = a2 x2 + 2abx + b2 (3)
If the two terms are ax + b and ax − b then their product is:
(ax + b) (ax − b) = a2 x2 − b2 (4)
This is known as the dierence of two squares.
2.3 Factorisation
Factorisation is the opposite of expanding brackets. For example expanding brackets would require 2 (x + 1)
to be written as 2x + 2. Factorisation would be to start with 2x + 2 and to end up with 2 (x + 1). In
previous grades, you factorised based on common factors and on dierence of squares.
http://cnx.org/content/m38375/1.3/
,OpenStax-CNX module: m38375 3
2.3.1 Common Factors
Factorising based on common factors relies on there being common factors between your terms. For example,
2x − 6x2 can be factorised as follows:
2x − 6x2 = 2x (1 − 3x) (5)
2.3.1.1 Investigation : Common Factors
Find the highest common factors of the following pairs of terms:
(a) 6y; 18x (b) 12mn; 8n (c) 3st; 4su (d) 18kl; 9kp (e) abc; ac
(f) 2xy; 4xyz (g) 3uv; 6u (h) 9xy; 15xz (i) 24xyz; 16yz (j) 3m; 45n
Table 2
2.3.2 Dierence of Two Squares
We have seen that:
(ax + b) (ax − b) = a2 x2 − b2 (6)
Since (6) is an equation, both sides are always equal. This means that an expression of the form:
a2 x2 − b2 (7)
can be factorised to
(ax + b) (ax − b) (8)
Therefore,
a2 x2 − b2 = (ax + b) (ax − b) (9)
For example, x2 −16 can be written as x2 − 42 which is a dierence of two squares. Therefore, the factors
of x2 − 16 are (x − 4) and (x + 4).
Exercise 2: Factorisation (Solution on p. 13.)
Factorise completely: b2 y 5 − 3aby 3
Exercise 3: Factorising binomials with a common bracket (Solution on p. 13.)
Factorise completely: 3a (a − 4) − 7 (a − 4)
Exercise 4: Factorising using a switch around in brackets (Solution on p. 13.)
Factorise 5 (a − 2) − b (2 − a)
http://cnx.org/content/m38375/1.3/
, OpenStax-CNX module: m38375 4
2.3.2.1 Recap
1. Find the products of:
(a) 2y (y + 4) (b) (y + 5) (y + 2) (c) (y + 2) (2y + 1)
(d) (y + 8) (y + 4) (e) (2y + 9) (3y + 1) (f) (3y − 2) (y + 6)
Table 3
Click here for the solution1
2. Factorise:
a. 2l + 2w
b. 12x + 32y
c. 6x2 + 2x + 10x3
d. 2xy 2 + xy 2 z + 3xy
e. −2ab2 − 4a2 b
Click here for the solution2
3. Factorise completely:
(a) 7a + 4 (b) 20a − 10 (c) 18ab − 3bc
(d) 12kj + 18kq (e) 16k2 − 4k (f) 3a2 + 6a − 18
(g) −6a − 24 (h) −2ab − 8a (i) 24kj − 16k2 j
(j) −a2 b − b2 a (k) 12k2 j + 24k2 j 2 (l) 72b2 q − 18b3 q 2
(m) 4 (y − 3) + k (3 − y) (n) a (a − 1) − 5 (a − 1) (o) bm (b + 4) − 6m (b + 4)
(p) a2 (a + 7) + a (a + 7) (q) 3b (b − 4) − 7 (4 − b) (r) a2 b2 c2 − 1
Table 4
Click here for the solution3
3 More Products
Khan Academy video on products of polynomials.
This media object is a Flash object. Please view or download it at
<http://www.youtube.com/v/fGThIRpWEE4&rel=0>
Figure 9
We have seen how to multiply two binomials in "Product of Two Binomials" (Section 2.2: Product of Two
Binomials). In this section, we learn how to multiply a binomial (expression with two terms) by a trinomial
1 http://www.fhsst.org/lxI
2 http://www.fhsst.org/lqV
3 http://www.fhsst.org/lqE
http://cnx.org/content/m38375/1.3/
