Easter School
Mathematics
Algebraic Expressions
Grade 10
01 April 2013
KEY CONCEPTS
Manipulate algebraic expressions by:





Multiplying a binomial by a trinomial;
Factorising trinomials;
Factorising the difference and sums of two cubes;
Factorising by grouping in pairs; and
Simplifying, adding and subtracting algebraic fractions with denominators of
cubes.
TERMINOLOGY
Binomial: an expression with two terms
Trinomial: an expression with three terms
Factorising: the opposite process of expanding the brackets
X-PLANATION
Products
When multiplying a binomial or trinomial by a single term we place the binomial or
trinomial in brackets. We use the Distributive Law of multiplication to expand term by
term. This means that we multiply each term inside a bracket by the term outside the
brackets.
Simplify:
Solution:
=
=
=
When multiplying two binomials together we apply the same Distributive Law. We
st
multiple the two First term together, then the Outer terms (1 term of the first bracket
nd
nd
by the 2 term of the second bracket) then the Inner terms (2 term of the first
st
bracket by the 1 term of the second bracket) and then Last terms. Remember FOIL
Find the product:
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Easter School
Mathematics
Algebraic Expressions
Grade 10
01 April 2013
Solution:
=
=
=
Multiplying a Binomial by Trinomial
Find the product:
Solution:
Step 1:
Expand the bracket
=
=
Step 2:
Simplify
=
Methods of Factorising





Taking out a common factor
Difference of two squares
Trinomials to binomials
Sum and difference of cubes
Grouping
Factorise:
Solution:
(
) and (
) are not common factors. The terms have different signs. If we
take out a factor of -1 from (2-a), we can change the signs:
Now we have a common factor in both terms. Be careful of the signs.
=
=
=
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Easter School
Mathematics
Algebraic Expressions
Grade 10
01 April 2013
Factorising Trinomials
Recognise the pattern. The process is the opposite of FOIL.
The factors of the first term of a trinomial become the first terms of each of the
binomials which we place in brackets.
The factors of the last term of a trinomial become the second term of each of the
binomials in the brackets.
We select the factors so that when we expand the multiplication of inner and outer
terms gives us the middle term of the trinomial.
Always check that you have the correct factors by expanding the factors using FOIL.
Difference and sum of two cubes
You need to recognise the pattern. Notice that the product of a binomial and trinomial
gives us the sum of two cubes.
=
=
=
=
Also this product gives the difference of two cubes:
=
=
=
=
So to factorise the sum of cubes, we use the pattern. The terms in the first bracket
are added together and the middle term in the second bracket is subtracted.
=
For the difference of cubes, the terms in the first bracket are subtracted and the
middle term in the second bracket is added.
=
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Easter School
Mathematics
Algebraic Expressions
Grade 10
01 April 2013
Simplifying Algebraic Fractions
We working with algebraic fractions in the same way we work with fractions of
numbers.
Multiplication and division
Simplify:
, ( ≠ 0;
≠ ±2)
Solution:
Step 1:
Factorise the numerator and denominator
Step 2:
Change the division sign and multiply by the reciprocal
Step 3:
Write the final answer
=1
Addition and Subtraction
Question
Simplify:
–
+
Step 1:
, (x ≠ ±2)
Factorise the denominators
–
+
Step 2:
Make all denominators the same so that we can add or subtract
the fractions
The lowest common denominator is
+
–
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Easter School
Mathematics
Algebraic Expressions
Grade 10
01 April 2013
Step 3:
Write as one fraction
Step 4:
Simplify
Step 5:
Take out the common factor and write the final answer
X-AMPLE QUESTIONS:
Question 1
Simplify the following:
a.)
b.)
c.)
d.)
2
(a - 2) - a(a + 4)
2
2
(5a - 4b)(25a + 20ab + 16b )
2
(2m - 3)(4m + 9)(2m +3)
(a + 2b - c)(a + 2b + c)
Question 2
Factorise the following trinomials
a.)
b.)
c.)
d.)
2
x + 8x + 15
2
x - 4x - 12
2
x + 4x - 12
2
x - 6x + 9
Question 3
Factorise the following expressions fully:
4
a.) (16 - x )
2
b.) (m - )
c.)
d.)
e.)
f.)
g.)
2
9b - 81
2
16b + 56b + 49
2
2
2a - 12ab + 18b
2
3
y - 7y - 30 g) 3p – 1/9
3
3
(2 + p) - 8(p + 1)
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Easter School
Mathematics
Algebraic Expressions
Grade 10
01 April 2013
Question 4
Simplify the following fractions:
a.)
x
b.)
c.)
d.)
Question 5
2
2
Show that (2x -1) - (x - 3) can be simplified to (x + 2)(3x -4)
Question 6
2
2
What must be added to x - x + 4 to make it equal to (x + 2) ?
Question 7
Evaluate
If
= 7,85 without using a calculator. Show your working.
Question 8
3
3
With what expression must (a - 2b) be multiplied to get a product of a - 8b ?
Question 9
3
With what expression must 27x + 1 be divided to get a quotient of 3x + 1?
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