© boardworks ltd 2005 1 of 73 a1 algebraic manipulation ks4 mathematics
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© Boardworks Ltd 2005 2 of 73
Contents
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AA1.1 Using index laws
A1 Algebraic manipulation
A1.2 Multiplying out brackets
A1.3 Factorization
A1.5 Algebraic fractions
A1.4 Factorizing quadratic expressions
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Multiplying terms
Simplify:
x + x + x + x + x = 5x
Simplify:
x × x × x × x × x = x5
x to the power of 5
x5 as been written using index notation.
xn
The number x is called the base.
The number n is called the index or power.
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We can use index notation to simplify expressions.
For example,
3p × 2p = 3 × p × 2 × p = 6p2
q2 × q3 = q × q × q × q × q = q5
3r × r2 = 3 × r × r × r = 3r3
3t × 3t = (3t)2 or 9t2
Multiplying terms involving indices
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Multiplying terms with the same base
For example,
a4 × a2 = (a × a × a × a) × (a × a)
= a × a × a × a × a × a
= a6
When we multiply two terms with the same base the indices are added.When we multiply two terms with the same base the indices are added.
= a (4 + 2)
In general,
xm × xn = x(m + n)xm × xn = x(m + n)
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Dividing terms
Remember, in algebra we do not usually use the division sign, ÷.
Instead, we write the number or term we are dividing by underneath like a fraction.
For example,
(a + b) ÷ c is written as a + bc
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Like a fraction, we can often simplify expressions by cancelling.
For example,
n3 ÷ n2 =n3
n2
=n × n × n
n × n
= n
6p2 ÷ 3p =6p2
3p
=6 × p × p
3 × p
2
= 2p
Dividing terms
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Dividing terms with the same base
For example,
a5 ÷ a2 =a × a × a × a × a
a × a= a × a × a = a3
4p6 ÷ 2p4 =4 × p × p × p × p × p × p
2 × p × p × p × p= 2 × p × p = 2p2
= a (5 – 2)
= 2p(6 – 4)
When we divide two terms with the same base the indices are subtracted.When we divide two terms with the same base the indices are subtracted.
In general,
xm ÷ xn = x(m – n)xm ÷ xn = x(m – n)
2
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Sometimes terms can be raised to a power and the result raised to another power.
For example,
(y3)2 = (pq2)4 =
Expressions of the form (xm)n
y3 × y3
= (y × y × y) × (y × y × y)
= y6
pq2 × pq2 × pq2 × pq2
= p4 × q (2 + 2 + 2 + 2)
= p4 × q8
= p4q8
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Expressions of the form (xm)n
For example,
(a5)3 = a5 × a5 × a5
= a(5 + 5 + 5)
= a15
When a term is raised to a power and the result raised to another power, the powers are multiplied.When a term is raised to a power and the result raised to another power, the powers are multiplied.
= a(3 × 5)
In general,
(xm)n = xmn(xm)n = xmn
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Expressions of the form (xm)n
Rewrite the following without brackets.
1) (2a2)3 = 8a6 2) (m3n)4 = m12n4
3) (t–4)2 = t–8 4) (3g5)3 = 27g15
5) (ab–2)–2 = a–2b4 6) (p2q–5)–1 = p–2q5
7) (h½)2 = h 8) (7a4b–3)0 = 1
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The zero index
Look at the following division:
y4 ÷ y4 = 1
But using the rule that xm ÷ xn = x(m – n)
y4 ÷ y4 = y(4 – 4) = y0
That means that
y0 = 1
In general, for all x 0,
x0 = 1x0 = 1
Any number or term divided by itself is equal to 1.
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Negative indices
Look at the following division:
b2 ÷ b4 =b × b
b × b × b × b=
1b × b
=1b2
But using the rule that xm ÷ xn = x(m – n)
b2 ÷ b4 = b(2 – 4) = b–2
That means that
b–2 = 1b2
In general,
x–n = 1xn
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Negative indices
Write the following using fraction notation:
u–1 = 1u
2b–4 = 2b4
x2y–3 = x2
y3
This is the reciprocal of u.
2a(3 – b)–2 = 2a
(3 – b)2
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Negative indices
Write the following using negative indices:
2a
=
x3
y4=
p2
q + 2=
3m(n2 + 2)3
=
2a–1
x3y–4
p2(q + 2)–1
3m(n2 + 2)–3
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Indices can also be fractional.
Fractional indices
x × x =12
12 x + =
12
12 x1 = x
But, x × x = x
x1 = x
So, x = x x = x 12
Similarly, x × x × x =13
13
13 x + + =
13
13
13
But, x × x × x = x3 3 3
So, x = x x = x 13 3
The square root of x.
The cube root of x.
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x = x x = x
In general,
Fractional indices
Also, we can write x as x . mn
1n × m
Using the rule that (xm)n = xmn, we can write
1n n
We can also write x as xm × . mn
1n
x × m = (x )m = (x)m1n
1n n
In general,
x = xmx = xm x = (x)mx = (x)mmn n or
mn n
x = (xm) = xm1nm×
n1n
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Here is a summary of the index laws.
xm × xn = x(m + n)
xm ÷ xn = x(m – n)
Index laws
(xm)n = xmn
x1 = x
x0 = 1 (for x = 0)
x = x 1n
n
x = x 12
x = xm or (x)mnmn n
x–1 = 1x
x–n = 1xn
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Contents
A
A
A
A
A
A1.2 Multiplying out brackets
A1.3 Factorization
A1.1 Using index laws
A1 Algebraic manipulation
A1.5 Algebraic fractions
A1.4 Factorizing quadratic expressions
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Look at this algebraic expression:
Expanding expressions with brackets
3y(4 – 2y)
This means 3y × (4 – 2y), but we do not usually write × in algebra.
To expand or multiply out this expression we multiply every term inside the bracket by the term outside the bracket.
3y(4 – 2y) = 12y – 6y2
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Look at this algebraic expression:
Expanding expressions with brackets
–a(2a2 – 2a + 3)
When there is a negative term outside the bracket, the signs of the multiplied terms change.
–a(2a2 – 3a + 1) = –2a3 + 3a2 – a
In general, –x(y + z) = –xy – xz
–x(y – z) = –xy + xz
–(y + z) = –y – z
–(y – z) = –y + z
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Expanding brackets and simplifying
Sometimes we need to multiply out brackets and then simplify.
For example, 3x + 2x(5 – x)
We need to multiply the bracket by 2x and collect together like terms.
3x + 10x – 2x2
= 13x – 2x2
3x + 2x(5 – x) =
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Expanding brackets and simplifying
Expand and simplify: 4 – (5n – 3)
We need to multiply the bracket by –1 and collect together like terms.
4 – 5n + 3
= 4 + 3 – 5n
= 7 – 5n
4 – (5n – 3) =
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Expanding brackets and simplifying
Expand and simplify: 2(3n – 4) + 3(3n + 5)
We need to multiply out both brackets and collect together like terms.
6n – 8 + 9n + 15
= 6n + 9n – 8 + 15
= 15n + 7
2(3n – 4) + 3(3n + 5) =
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We need to multiply out both brackets and collect together like terms.
15a + 10b – 2a – 5ab
= 15a – 2a + 10b – 5ab
= 13a + 10b – 5ab
Expanding brackets then simplifying
5(3a + 2b) – a(2 + 5b) =
Expand and simplify: 5(3a + 2b) – a(2 + 5b)
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Find the area of the rectangle
What is the area of a rectangle of length (a + b) and width (c + d)?
a b
c
d
ac bc
ad bd
In general,(a + b)(c + d) = ac + ad + bc + bd
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Expanding two brackets
Look at this algebraic expression:
(3 + t)(4 – 2t)
This means (3 + t) × (4 – 2t), but we do not usually write × in algebra.
To expand or multiply out this expression we multiply every term in the second bracket by every term in the first bracket.
(3 + t)(4 – 2t) = 3(4 – 2t) + t(4 – 2t)
= 12 – 6t + 4t – 2t2
= 12 – 2t – 2t2
This is a quadratic
expression.
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Expanding two brackets
With practice we can expand the product of two linear expressions in fewer steps. For example,
(x – 5)(x + 2) = x2 + 2x – 5x – 10
= x2 – 3x – 10
Notice that –3 is the sum of –5 and 2 …
… and that –10 is the product of –5 and 2.
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Squaring expressions
Expand and simplify: (2 – 3a)2
We can write this as,
(2 – 3a)2 = (2 – 3a)(2 – 3a)
Expanding,
(2 – 3a)(2 – 3a) = 2(2 – 3a) – 3a(2 – 3a)
= 4 – 6a – 6a + 9a2
= 4 – 12a + 9a2
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Squaring expressions
In general,
(a + b)2 = a2 + 2ab + b2
The first term squared …
… plus 2 × the product of the two terms …
… plus the second term squared.
For example,
(3m + 2n)2 = 9m2 + 12mn + 4n2
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The difference between two squares
Expand and simplify (2a + 7)(2a – 7)
Expanding,
(2a + 7)(2a – 7) = 2a(2a – 7) + 7(2a – 7)
= 4a2 – 14a + 14a – 49
= 4a2 – 49
When we simplify, the two middle terms cancel out.
In general,
(a + b)(a – b) = a2 – b2 (a + b)(a – b) = a2 – b2
This is the difference between two squares.
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Contents
A
A
A
A
A
A1.3 Factorization
A1.2 Multiplying out brackets
A1.1 Using index laws
A1 Algebraic manipulation
A1.5 Algebraic fractions
A1.4 Factorizing quadratic expressions
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Factorizing expressions
Factorizing an expression is the opposite of expanding it.
a(b + c) ab + ac
Expanding or multiplying out
FactorizingOften:When we expand an expression we remove the brackets.When we factorize an expression we write it with brackets.
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Factorizing expressions
Expressions can be factorized by dividing each term by a common factor and writing this outside of a pair of brackets.
For example, in the expression
5x + 10
the terms 5x and 10 have a common factor, 5.
We can write the 5 outside of a set of brackets
5(x + 2)
We can write the 5 outside of a set of brackets and mentally divide 5x + 10 by 5.
(5x + 10) ÷ 5 = x + 2
This is written inside the bracket.
5(x + 2)
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Factorizing expressions
Writing 5x + 10 as 5(x + 2) is called factorizing the expression.
Factorize 6a + 8
6a + 8 = 2(3a + 4)
Factorize 12n – 9n2
12n – 9n2 = 3n(4 – 3n)
The highest common factor of 6a and 8 is 2.
(6a + 8) ÷ 2 = 3a + 4
The highest common factor of 12n and 9n2 is 3n.
(12n – 9n2) ÷ 3n = 4 – 3n
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Factorizing expressions
Writing 5x + 10 as 5(x + 2) is called factorizing the expression.
3x + x2 = x(3 + x)2p + 6p2 – 4p3
= 2p(1 + 3p – 2p2)
The highest common factor of 3x and x2 is x.
(3x + x2) ÷ x = 3 + x
The highest common factor of 2p, 6p2 and 4p3 is 2p.
(2p + 6p2 – 4p3) ÷ 2p
= 1 + 3p – 2p2
Factorize 3x + x2 Factorize 2p + 6p2 – 4p3
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Factorization by pairing
Some expressions containing four terms can be factorized by regrouping the terms into pairs that share a common factor. For example,
Factorize 4a + ab + 4 + b
Two terms share a common factor of 4 and the remaining two terms share a common factor of b.
4a + ab + 4 + b = 4a + 4 + ab + b
= 4(a + 1) + b(a + 1)
4(a + 1) and + b(a + 1) share a common factor of (a + 1) so we can write this as
(a + 1)(4 + b)
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Factorization by pairing
Factorize xy – 6 + 2y – 3x
We can regroup the terms in this expression into two pairs of terms that share a common factor.
xy – 6 + 2y – 3x = xy + 2y – 3x – 6
= y(x + 2) – 3(x + 2)
y(x + 2) and – 3(x + 2) share a common factor of (x + 2) so we can write this as
(x + 2)(y – 3)
When we take out a factor of
–3, – 6 becomes + 2
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Contents
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A1.4 Factorizing quadratic expressions
A1.3 Factorization
A1.2 Multiplying out brackets
A1.1 Using index laws
A1 Algebraic manipulation
A1.5 Algebraic fractions
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Quadratic expressions
A quadratic expression is an expression in which the highest power of the variable is 2. For example,
x2 – 2, w2 + 3w + 1, 4 – 5g2 ,t2
2The general form of a quadratic expression in x is:
x is a variable.
a is a fixed number and is the coefficient of x2.
b is a fixed number and is the coefficient of x.
c is a fixed number and is a constant term.
ax2 + bx + c (where a = 0)
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Factorizing expressions
Remember: factorizing an expression is the opposite of expanding it.
Expanding or multiplying out
FactorizingOften:When we expand an expression we remove the brackets.
(a + 1)(a + 2) a2 + 3a + 2
When we factorize an expression we write it with brackets.
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Factorizing quadratic expressions
Quadratic expressions of the form x2 + bx + c can be factorized if they can be written using brackets as
(x + d)(x + e)
where d and e are integers.
If we expand (x + d)(x + e) we have,
(x + d)(x + e) = x2 + dx + ex + de
= x2 + (d + e)x + de
Comparing this to x2 + bx + c we can see that:
The sum of d and e must be equal to b, the coefficient of x.
The product of d and e must be equal to c, the constant term.
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Factorizing quadratic expressions
Quadratic expressions of the form ax2 + bx + c can be factorized if they can be written using brackets as
(dx + e)(fx + g)
where d, e, f and g are integers.
If we expand (dx + e)(fx + g)we have,
(dx + e)(fx + g)= dfx2 + dgx + efx + eg
= dfx2 + (dg + ef)x + eg
Comparing this to ax2 + bx + c we can see that we must choose d, e, f and g such that: a = df,
b = (dg + ef)
c = eg
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Factorizing the difference between two squares
A quadratic expression in the form
x2 – a2
is called the difference between two squares.
The difference between two squares can be factorized as follows:
x2 – a2 = (x + a)(x – a)
For example,
9x2 – 16 = (3x + 4)(3x – 4)
25a2 – 1 = (5a + 1)(5a – 1)
m4 – 49n2 = (m2 + 7n)(m2 – 7n)
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Contents
A
A
A
A
A
A1.5 Algebraic fractions
A1.3 Factorization
A1.2 Multiplying out brackets
A1.1 Using index laws
A1 Algebraic manipulation
A1.4 Factorizing quadratic expressions
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Algebraic fractions
The rules that apply to numerical fractions also apply to algebraic fractions.
For example, if we multiply or divide the numerator and the denominator of a fraction by the same number or term we produce an equivalent fraction.
3x4x2
and are examples of algebraic fractions.2a
3a + 2
For example,
3x4x2
=34x
=68x
=3y4xy
=3(a + 2)4x(a + 2)
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Simplifying algebraic fractions
We simplify or cancel algebraic fractions in the same way as numerical fractions, by dividing the numerator and the denominator by common factors. For example,
Simplify 6ab3ab2
6ab3ab2
=6 × a × b
3 × a × b × b
2
=2b
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Simplifying algebraic fractions
Sometimes we need to factorize the numerator and the denominator before we can simplify an algebraic fraction. For example,
Simplify 2a + a2
8 + 4a
=a4
2a + a2
8 + 4a=
a (2 + a)4(2 + a)
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Simplifying algebraic fractions
Simplify b2 – 363b – 18
b2 – 36 is the difference
between two squares.
b2 – 363b – 18
=(b + 6)(b – 6)
3(b – 6)
b + 63
=
If required, we can write this as
63
=b3
+b3
+ 2
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Manipulating algebraic fractions
Remember, a fraction written in the form
a + bc
can be written asbc
ac
+
However, a fraction written in the form
ca + b
cannot be written ascb
ca
+
For example,
1 + 23
=23
13
+ but3
1 + 2=
32
31
+
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Multiplying and dividing algebraic fractions
We can multiply and divide algebraic fractions using the same rules that we use for numerical fractions.
In general, ab
× =cd
acbd
ab
÷ =cd
ab
× =dc
adbc
and,
For example,3p4
× =2
(1 – p)6p
4(1 – p)=
3
2
3p2(1 – p)
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23y – 6
÷ =4
y – 2
This is the reciprocal
of4
y – 2
23y – 6
×4
y – 2
23(y – 2)
×=4
y – 2
16
=
Multiplying and dividing algebraic fractions
2
What is2
3y – 6 ÷
4y – 2
?
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Adding algebraic fractions
We can add algebraic fractions using the same method that we use for numerical fractions. For example,
What is1a
+2b
?
We need to write the fractions over a common denominator before we can add them.
1a
+2b
=b + 2a
abbab
+2aab
=
In general,
+ =ab
cd
ad + bcbd
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Adding algebraic fractions
What is3x
+y2
?
We need to write the fractions over a common denominator before we can add them.
3x
+y2
=
=6 + xy
2x
+62x
xy2x
=
+3 × 2x × 2
y × x2 × x
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Subtracting algebraic fractions
We can also subtract algebraic fractions using the same method as we use for numerical fractions. For example,
We need to write the fractions over a common denominator before we can subtract them.
In general,
What is – ?p3
q2
– =p3
q2
– =2p6
3q6
2p – 3q6
– =ab
cd
ad – bcbd
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Subtracting algebraic fractions
What is – ?
–(2 + p) × 2q
4 × 2q3 × 42q × 4
2 + p4
32q
=–2 + p
432q
= –2q(2 + p)
8q128q
=2q(2 + p) – 12
8q4
6
=q(2 + p) – 6
4q