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UNIT 5. EXPRESSIONS USING ALGEBRA
5. 1 ALGEBRAIC EXPRESSIONS
Objective
Learn how to replace numbers with letters called variables.
Summary
Algebra is an area of mathematics where letters are used to represent numbers.
You can use algebra to solve mathematical problems.
A variable is a letter used to represent an unknown number, for example, x.
In algebra, letters are used when numbers are not known, unknown.
r + 2s means an unknown number 'r', plus 2 lots of an unknown number 's'.
Q1. Say that 'g' is the cost of child admission, and 'k' is the cost of adult admission to
the zoo.
a) How much does it cost for the Khan family of 3 children and 3 adults to visit the zoo?
b) Write an algebraic expression for the cost for the Norman family of 5 children and 4
adults to visit the zoo.
http://kent.skoool.co.uk/content/keystage3/maths/pc/learningsteps/VRBLC/launch.html
(1)
1. Write variable expressions to represent word problems:
1. Annie has m buttons. Trey has
36 more buttons than Annie.
How many buttons Trey has?
2. Katrina has w hair bows.
Katrina's sister has 7 more hair
bows than Katrina. How many
hair bows Katrina's sister has?
3. Hannah had t books. Then she
went to a book bought 89 more
books. How many books Hannah
has now?
4. Jeremy has z trading cards.
Martha has 37 more trading
cards than Jeremy. How many
trading cards Martha has?
5. There were 5 frogs in a pond. k
of the frogs hopped away. How
many frogs are in the pond now?
1
6. Colin has m small spoons and
25 big spoons. Write an
expression that shows how many
spoons Colin has.
7. Emilie has w trading cards. Tori
has 61 more trading cards than
Emilie. Write an expression that
shows how many trading cards
Tori has.
8. Mason planted 94 fewer trees
than Larry. Larry planted k
trees. Write an expression that
shows how many trees Mason
planted.
9. Meredith has 40 buttons. She
gives away p buttons. Write an
expression that shows the
number of buttons Meredith has
left.
shows how many apples Wyatt
picked.
12. Alex had 6 CDs until he won r
more in a contest. How many
CDs Alex has now?
13. The advert says that the cost of a
party is 50€ plus 5€ per person,
a) So what would be the cost of a
party for 10?
b) So what would be the cost of a
party for 20?
c) How do you work out the cost if
you don't know the number of
people who will attend the
party?
14. Alexa is 12 years old. How old
will she be in n years time?
10. Jake ate u out of 83 gumdrops.
Write an expression that shows
how many gumdrops Jake has
left.
15. Roland weighs 70 kilograms,
and Mark weighs k kilograms.
Write an expression for their
combined weight.
11. Nicole picked 97 apples. Wyatt
picked b fewer apples than
Nicole. Write an expression that
16. A car travels down the freeway
at 55 km/h. Write an expression
for the distance the car will have
travelled after h hours.
http://kent.skoool.co.uk/keystage3.aspx?id=65 > Multiplication 27 > Scene 1 – 4
(2)
Algebraic terms, like 2s or 8y, leave the multiplication signs out. So rather than '2 · s',
write 2s, rather than '8 ·× y' write 8y.
A string of numbers and letters joined together by mathematical operations such as +
and - is called an algebraic expression
2. Write using algebraic expressions:
a) The double of a number plus 5 units.
d) The cube of a number.
b) The triple of a number less 6 units.
e) A number plus its square.
c) The square of a number.
f) Half of a number minus 3.
2
Solving problems in algebra depends on your ability to represent
missing or unknown quantities. Representing unknown quantities
is easy to do if you know the "language".
For example, what operation is meant by the phrase more than ?
If you guessed addition, then you are right! This skill of "translating" between
words and mathematical operations just takes a little vocabulary drill. Below
are some of the most common phrases used in problem solving, together with
the operations they represent. Study these relationships and you should be a
whiz at algebraic representation.
ADD
add
sum
more than
increased by
exceeds
in all
total
plus
SUBTRACT
subtract
difference
*less than
decreased by
diminished by
minus
fewer
reduced by
MULTIPLY
multiplied by
of
product
times
double
twice
triple
DIVIDE
divide
quotient
divided equally
per
ratio of
* be careful using "less than" - it reverses the order of things
Also, be careful of the placement of commas in statements.
In the statement "the sum of a and b, divided by 3" the comma indicates that
the answer is (a+b)/3 and not a + b/3.
EXAMPLES
1.
two more than a number
2+x
2.
five less than three times a number
3x - 5
notice how this changed the order
3.
seven times a number, increased by 4
4.
six decreased by 5 times a number
5.
7x + 4
6 - 5x
Given 2x - 4, write a verbal expression that
matches this mathematical expression.
Some possible verbal answers:
twice a number decreased by 4
double a number minus
3
Numerical value of an algebraic expression
1. What is the value of 3(2x - 5) when x = 4?
a) 6x – 5
b) 9
c) 30
2. What is the value of 4(y + 10) when y= 5?
a) y + 40
b) 40
c) 60
3. What is the value of 6g + 10k when g = 5 and k = 2?
a) 31
b) 40
c) 50
4. What is the value of 2x2- y when:
a) x=0, y =1
b)x=-1, y=-2
5. What is the value of x2 + 1 when x=2?
6. Calculate the following expressions:

a + bc2 − c(b + ac)2 =
a=2
b=3
c = (-3)
 3n2 + 4n =
n=1

b − (ac)2 + (3 − b) =
a=4
b = (-2)
c=3
 n(n+3) =
n = -2

n2 (5+ 4n) =
 n3 + n2 + n =
n = -1
n=2
4
5.2 Monomials
In algebra, a letter can be used to represent a number that you do not know. This gives
us algebraic terms, like 2x and 5y, called monomials.
Monomial is a product of two or some factors, each of them is either a number, or a
letter, or a power of a letter. For example,
3 a 2 b 4 , b d 3 , – 17 a b c
are monomials. A single number or a single letter may be also considered as a
monomial. The numerical factor is called a coefficient.
Degree of monomial is a sum of exponents of the powers of all its letters.
Q1. Complete:
Monomial
-8xy2
3a2b3
Coefficient
Letters
4
-9
2
3
x3y2
abc
Degree
b2
When algebraic terms are put together by mathematical operations such as + and - we
get an algebraic expression, like 2x + 5y +3z.
We can simplify algebraic expressions by collecting 'like terms'.
Like terms
Monomials are called similar or like terms if they are identical or differed only by
coefficients.
-25a2 and 8a2 are like terms.
3z2 and -4z are not like terms
Monomials that have the same letter are called like terms (monomis semblants). Only
like terms can be added (+) or subtracted (-).
Examples:
Terms
7x
Why are they "Like Terms"
x
-2x
(1/3)xy2 -2xy2 6xy2
because the variables are all x
because the variables are all xy2
5
Q2. Write a like term in each monomial from Q1.
To simplify an expression, we collect like terms. Like terms include letters and
numbers that are the same.
Look at the expression 4x + 5x -2 - 2x + 7
To simplify:


The x terms can be collected together to give 7x.
The numbers can be collected together to give 5.
So 4x + 5x -2 - 2x + 7 simplified is 7x + 5.
Remember that terms are separated by + and - signs, and these are always attached to
the front of a term.
Example
So 9b, -7b and 13b are like terms, but 6t, 5x and -11z are not like terms.
When like terms are added and subtracted it is called simplifying.
Different terms
To answer some exam questions you will have to simplify an expression that has many
different terms or letters.
Have a look at this typical exam question. You will notice that there are three different
terms in this question: x, y and z.
Q3. Simplify the expressions 5x + 3y - 6x + 4y + 3z =
5a + 3b - 3a - 5c + 4b =
a)
b)
c)
d)
e)
f)
2 x 2  3x 2 
8 x 3  14 x 3 
6x9  4x9 
12 x 5  4 x 5 
11x 4  7 x 4 
13x 6  8 x 7 
2 3 1 3
x  x 
3
2
5
h) x  2 x5 
1
i) y 2  2 y 2  y 2 
2
1
g) 3 y 7  y 7  2 y 7 
3
g)
6
5.3 Operating with monomials
5.3.1 Adding and subtracting like terms
The Khan family and the Norman family visit the zoo together - there are 8 children and
7 adults in the group. Because there are more than 10 people, the families can take
advantage of a special offer (1 child can be admitted free of charge).
As before let's use g for the cost of child admission, and k for the cost of adult
admission.
Cost for the Khan family = 3g + 3k
Cost for the Norman family = 5g + 4k
Offer = - g
Total cost = 3g + 3k + 5g + 4k - g
=
7g + 7k
http://kent.skoool.co.uk/keystage3.aspx?id=65#24_30 > 25. Terms (3)
1. Simplify these expressions:
1) a+a+a =
2) 2b+3b =
3) a+b+a+b+c =
4) 6c+2d+3c+3d
+ and – signs
A term, like a number, belongs with the sign that sits in front of it. So in the expression
2k - g, g belongs with the - sign that sits in front of it, so it is - g; and 2k belongs with
the + sign that sits in front of it, making it + 2k.
Remember:


Collect all the like terms together, eg, re-write the expression 3g + 2k + 5g + 4k
- g with all the g s and all the k s together: 3g + 5g - g + 2k + 4k
When you add or subtract terms, keep each term with their + or - sign.
In algebra, we often get very long expressions that we need to make simpler. Simpler
expressions are easier to solve.
http://kent.skoool.co.uk/keystage3.aspx?id=65 > 26. Expressions (4)
7
Questions
1. Simplify these expressions:
a) x + 5 + 3x- 7 + 9x+ 3 - 4x =
b) 4x + y - 2x + 6y =
c) 3g + 2k + 5g + 4k - g =
2. When the expression 3a + 9b - 2b is simplified, it becomes:
a) 12ab - 2b
b) 3a + 7b
c) 3a + 11b
3. When the expression 14m - 6k - 5m + 3k + 4k is simplified, it becomes:
a) 9m + k
b) 9m + 13k
c) 19m - 7k
Remember: Like terms with powers can be added or subtracted but only if the powers
are the same. x is not the same as x2, so they cannot be added together.
Example
Rearrange the expression so that like
terms are next to each other
Simplify:
3x + 5 + x2 - 2x + 2x2 - 1 = x2 + 2x2 + 3x - 2x + 5 – 1 = 3x2 + x + 4
Simplify
People usually write the expression with the 'x2' term first, then the 'x' term and finally
the number term. In general, in decreasing order.
Questions
4. Simplify:
a)
b)
c)
d)
e)
f)
g)
5 + 4x2 - 3 + 2x + x2 - 4x =
5x + 2x =
−3y2 + 4y2 =
2ab2 − a2b =
5x2 + 7x =
5x3 −6x + 7x − x3 −x + 4x3=
−x2 + x + x2 + x3 + x =
8
h)
i)
j)
k)
2x3 −(x3 −3x3) =
8x2 −x + 9x + x2 =
8xy2 −5x2y + x2y −xy2=
−3x + 7y −(8y + y −6x) =
5. Write true or false:
a) a + a = 2a
b) 2a + a = 2 a2
c) 2a − a =2
d) 2a −2 = a
e) 2a −b = 2 · (a −b)
f) 2a + 3a = 5a
g) 2a + 3b = 5ab
h) 2a2 = 4a
6. Complete :
a) (6x2 −4x + 7) + (8x2 +2x-3) =
b) (5x3 + 3x2 −10) − (6x2 −4x + 7) =
c) (9x3 + x2 − 6x + 4) + (x2 −4x + 3) - (6x2 −2x + 7) =
5.3.2 Multiplying and dividing monomials
Multiplying letters
Monomials can be multiplied in the same way as numbers.
a · a = a2,
b · b = b2 etc.
Remember that 2a is not the same as a2
2a = 2 · a
a2 = a · a
In general, am · an = a(m + n)
Questions
Simplify:
a) a2 · a3 =
b) p4 · p2 =
Remember that x is the same as x1 when using this rule. eg, n3 · n = n(3 + 1) = n4
9
Multiplying numbers and letters
Multiply letters and the numbers separately:
2 · 3a means 2 · 3 · a which is 6a
4a · 5a means 4 · a · 5 · a which is 4 · 5 · a · a = 20a2
Question
1. Simplify
b) 2x2· 4x3 ·5x6=
a)2p · 3p2 =
c) −3x ·(−2x)·
7x

4
d) 7x 3 ·5x · 9x4=
2.Complete:
·
3x
4x
5x 2
2x
6x 2
2x 2
Objective: Understand how to multiply variables and terms in an expression.
Summary
When multiplying terms, remember the following steps:
1. First, multiply the numbers.
2. Then multiply the variables.
3. Write the numeral first, followed by the variable.
Dividing letters
Algebraic expressions can be divided in the same way as numbers.
x8
 x5
3
x
x4
 x3
x
In general, a m : a n  a m  n
10
Dividing numbers and letters
Divide letters and the numbers separately:
6x5 6 x5
 · 2  3x 3
2
2 x
2x
6
12 x
12 x 6 4 x 3

· 
5
15 x 3 15 x 3
2 3
2 3
 7x y
7 x y
 7 xy 3

·

4x
4
x
4
http://kent.skoool.co.uk/keystage3.aspx?id=65#24_30 > Algebraic fractions (5)
Simplify:
a) 15x3 : 5x2 =
b) −8x3y2 : 2x2y =
c) 10x4 yz2 : 5xyz =
d) −9a : 3a
e) −3y2 : 4y2=
f) −10x3y2 : x2y =
g) 2ab2 :a2b=
h) 10x3 : 2xy2=
i) 2x2 · x3 · 3x5 : (−6x)=
j) 8x4 : (2x2 + 2x2) =
k) (5y 3 −2y 3) : (3xy2) =
l)15x3 : 5x2=
m) −8x3 y2 : 2x2y=
n) 10x4yz 2 : 5xyz=
5.4 Polynomials
A monomial or the sum of two or more monomials is called a polynomial.
monomial that makes up a polynomial is referred to as a term of the polynomial.
6x5 - 3x4 - x3 - 9x + 7
Each
CONSTANT TERM
TERMS
11
A term is a number, variable or the product of a number and variable(s).
Examples of terms are :
A coefficient is the numeric factor of your term.
Here are the coefficients of the terms listed above:
Term
Coefficient
3
5
2
A constant term is a term that contains only a number. In other words, there is no
variable in a constant term.
There are special names for polynomials with 1, 2 or 3 terms:
Degree of polynomial is the most of degrees of monomials, forming this polynomial.

Give the degree of the following polynomial: 2x5 – 5x3 – 10x + 9
This polynomial has four terms, including a fifth-degree term, a third-degree
term, a first-degree term, and a constant term.
This is a fifth-degree polynomial.

Give the degree of the following polynomial: 7x4 + 6x2 + x
This polynomial has three terms, including a fourth-degree term, a seconddegree term, and a first-degree term. There is no constant term.
This is a fourth-degree polynomial
Q1. Give the degree of the following polynomial:


6x5 - 3x4 - x3 - 9x + 7 
2x3 - 3x2 - 1 
12
Evaluation
"Evaluating" a polynomial is the same as evaluating anything else: you plug in the given
value of x, and figure out what y is supposed to be. For instance:

Evaluate P(x)= 2x3 – x2 – 4x + 2 at x = –3
I need to plug in "–3" for the "x", remembering to be careful with my
parentheses and the negatives:
P(-3) = 2(–3)3 – (–3)2 – 4(–3) + 2
= 2(–27) – (9) + 12 + 2
= –54 – 9 + 14
= –63 + 14 = -49
Always remember
-
to be careful with the minus signs!
If we see a variable standing alone (it has no coefficient, no number next to it)
then we assume that there is an invisible one (1) standing there: x2 = 1x2
Q2. Simplify like terms in these polynomials, order them in decreasing order and give
the degree in each polynomial.
a) P (x ) = 5x3 −x + 7x3 −x2 + 8x −2
b) Q (x ) = 12 + x2 + 7x −x4 −8 + 3x2
c) R (x ) = 9x −4x2 −6 −10x + 1
d) S (x ) = 4x2 −x3 + 4x3 −x5 + 8 −x2
Q3. Evaluate Q(x ) = 3x4 −2x3 + x2 −5, at x= -1
Q4. Find the value of a to get P (x ) = ax2 −3x + 5verifies P (2) = 3.
Q5. If P (x ) = 3x4 −2x3+ x2 −5, calculate:
a) P (1) + P(0) −P (−2)
b) 2 ·P (2) + 3·(P (−1))
http://www.youtube.com/watch?v=IDpnNnjFB1c
Adding and Subtracting Polynomials (6)
Q6. Given :
A (x) = 2x3 −3x2 + x −7
B (x) = x3 + 7x2 − 4x
C (x) = −2x2 + x − 5
Then calculate:
a) A (x) + B (x) + C (x)
c) A (x) − B (x)
b) B (x) + C (x)
d) A (x) −B (x) −C (x)
13
Multiplying polynomials
MONOMIAL · POLYNOMIAL
Tools for Dealing
with
Distributive Property
a(b + c) = ab + ac
Rules for
Exponents
Multiplying
Signed Numbers
(+) • (+) = (+)
(+) • (-) = (-)
(-) • (+) = (-)
(-) • (-) = (+)
Monomials
Example 1:
monomial • monomial
(4x3) • (3x2)
= (4 • 3) • (x3 • x2)
= 12 • x5 = 12x5
Notice that the factors were
regrouped and then multiplied.
Also, multiply powers with the
same base by adding the
exponents.
= x2 + 4x
This problem requires the
distributive property. You need to
multiply each term in the
parentheses by the monomial
(distribute the x across the
parentheses).
Example 2:
monomial • binomial
Notice the distributive property at
work again.
Example 3:
monomial • trinomial
= 2x3 + 6x2 + 8x
Again, the distributive property is
needed along with the rule for
multiplying powers.
Example 4:
monomial •
polynomial
= 3x5 - 9x4 + 18x3 - 15x2
Multiply
–3x·(4x2 – x + 10) = –3x(4x2) – 3x(–x) – 3x(10) = –12x3 + 3x2 – 30x
x3(x4 + 5a) = x3(x4 + 5a) = x7 + 5ax3


2 x3 5x5  2 x 4  4 x3  6 x 2  x  7 
14
POLYNOMIAL· POLYNOMIAL
Multiply (x + 3)(x + 2)
1. "Distributive" Method:
The most universal method. Applies to all
polynomial multiplications, not just to binomials.
Start with the first term in the first binomial - the circled blue X. Multiply (distribute) this
term times EACH of the terms in the second binomial.
Now, take the second term in the first binomial - the circled red +3 (notice we take the sign
also). Multiply this term times EACH of the terms in the second binomial.
Add the results: x•x + x•2 + 3•x +3•2
x² + 2x +3x + 6 = x² + 5x + 6
(x + 5)(a − 6) = x(a − 6) + 5(a − 6) = ax − 6x + 5a − 30
(x − 3)2 = (x − 3)(x − 3) = x(x − 3) − 3(x − 3) = x2 − 3x − 3x + 9 = x2 − 6x + 9
(2x + 3)(x2 − x − 5) = (2x)(x2 − x − 5) + (3)(x2 − x − 5) =
= (2x3 − 2x2 − 10x) + (3x2 − 3x − 15)
= 2x3 + x2 − 13x − 15
We take the 2 terms of the first bracket and multiply both of them by the second bracket.
15
(x + 3)(x² + 2x + 4)
= x³ + 2x² + 4x + 3x² + 6x + 12
= x³ + 2x² + 3x² + 4x + 6x + 12 Group like terms.
= x³ + 5x² + 10x + 12
Q6. Multiply:
a) (3x + 4) ·2=
Combine like terms.
c) (4x 2 + x −2) · (−5) =
b) (x −2) · 4x =
d) (x2+ 3x −6) · (−3x3) =
Q7. Multiply and simplify:
a) (x + 3) · (x −2) =
b) (2x −6) · (3x + 5) =
c) (4 −6x + 3x2) · (−2 −x + x2 ) =
d) (x −5)2 =
e) (4 + a)2 =
f) (2x + 3y)2 =



g) x 5  x · x 2  x 



h) x 4  2 x 3  1 · x 3  5 
i)
(3x3  5x 2  3x  1)(3x3  5x)
j)
(5x5  3x  4)(7 x3  2 x  1)
k) (25x7  5x  2)( x 4  3x 2  3)
l)
( x3  2 x 2  2)(3x 4  5 x3  4 x 2  4)
m)  x  1 ( x  2)
2
n) ( x  3)  2 x 2  3x  1
2
Solutions:
9 x 6  15 x5  24 x 4  22 x3  15 x 2  5 x
8
6
5
4
3
2
j) 35 x  10 x  5 x  21x  28 x  6 x  5 x  4
11
9
7
5
4
3
2
k) 25 x  75 x  75 x  5 x  2 x  15 x  6 x  15 x  6
i)
16
3x 7  x 6  14 x5  14 x 4  14 x3  16 x 2  8
3
2
m) x  4 x  5 x  2
5
4
3
2
n) 4 x  24 x  41x  9 x  17 x  3
l)
Q8. Roll the two cubes, record the polynomials on the answer sheet, then
MULTIPLY the polynomials.
Q9. Practice with Multiplying Polynomials
http://www.regentsprep.org/Regents/math/ALGEBRA/AV3/PracPoly.htm
Special Pattern Binomials
The following are special multiplications involving binomials that you will want to try
to remember. Be sure to notice the patterns in each situation. You will be seeing these
patterns in numerous problems.
Don't panic! If you cannot remember these patterns, you can arrive at
your answer by simply multiplying with the distributive method. These
patterns are, however, very popular. If you can remember the patterns, you
can save yourself some work.
Squaring a Binomial - multiplying times itself
(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
Notice the middle terms in both of these problems. In each problem, the middle
term is twice the multiplication of the terms used to create the binomial
expression.
Example 1:
Example 2:
(x + 3)² = (x + 3)(x + 3)
= x² + 3x + 3x + 9 Distributive method
= x² + 6x + 9
 Notice the middle term.
(x - 4)² = (x - 4)(x - 4)
= x² - 4x - 4x + 16 Distributive method
= x² - 8x + 16
 Again, notice the middle term.
17
Product of Sum and Difference
(notice that the binomials differ only by the sign between the terms)
(a + b)(a - b) = a² - b²
Notice that there appears to be no "middle" term to forma a trinomial answer,
as was seen in the problems above. When multiplication occurs, the values that
would form the middle term of a trinomial actually add to zero.
Example 3: (x + 3)(x - 3) = x² - 3x + 3x - 9
Distributive method
= x² - 9
*Notice how the middle term is zero.
Example 4:
(2x + 3y)(2x - 3y) = 4x² - 6xy + 6xy - 9y²
Distributive method
= 4x² - 9y²
* Again, notice how the middle term is zero.
Q10. Calculate using patterns showed above:
a) (a + 4)2 =
b) (2x + 3)2 =
c) (b + 12)2 =
d) (3x + 5y)2 =
e) (x + 8y2)2 =
f) (5ab + 3b2)2 =
g) (50 – x)2 =
h) (ab – 20)2 =
i) (5x – 11)2 =
j) (
1 3 42
a –a ) =
2
k) (2xy – 7x)2 =
18
l) (3x – 4y3)2 =
m) (3 + x) · (3 – x) =
n) (x4 – 3y) · (x4 + 3y) =
o) (4 + 3x) · (3x – 4)=
p) (6ab + 3b2) · (6ab – 3b2) =
q) (– 2x – 10) · (10 – 2x) =
5.5 Formulas
What is an Equation?
An equation says that two things are equal. It will have an equals sign "=" like this:
x
+
2
=
6
That equations says: what is on the left (x + 2) is equal to what is on the right (6)
So an equation is like a statement "this equals that"
What is a Formula?
A formula is a special type of equation that shows the relationship between different
variables.
http://www.bbc.co.uk/schools/gcsebitesize/maths/algebra/formulaeact.shtml (7)
We often use formulas without even noticing. For example, we might convert miles to
km by multiplying by 1.6, or find the circumference of a circle by multiplying pi by the
diameter.
km = 1.6 x miles is an example of a formula.
L =  · d (circumference of a circle)
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Question. A rectangle has a width of x and a length of 2x.
Write down a formula for the perimeter (P) in terms of x. P=
TEST. Using formulas:
1. What's the formula connecting speed, distance and time?
a) s = d·t
b) s = d/t
c) s = d + t
2. If F = ma, what's the value of F when m = 5 and a = 3?
a) 8
b) 15
c) 53
3. If A = 3b - c2, find the value of A when b = 4 and c = 2
a) 30
b) 16
c) 8
4. If a = b - c, then:
a) c = b – a
b) c = a + b
c) c = a - b
b)
c) t(v - u)
5. If v = u + at, then a =:
a)
5.6 Extracting common factors
http://kent.skoool.co.uk/keystage4.aspx?id=317 > Factors 1 (8)
In Maths we multiply into brackets as follows:


3x  5 x 2  2 x  4 15x 3  6 x 2  12 x
We can also carry out the reverse process (extracting common factors). An expression
can be simplified by extracting out the common factor.
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When factoring polynomials, first look for the largest monomial which is a factor of
each term of the polynomial. Factor out (divide each term by) this largest monomial.
15 x 3  6 x 2  12 x = 5·3·x·x·x - 2·3·x·x + 4·3·x =
= 5·3·x·x·x - 2·3·x x + 4·3·x =3x · 5x 2  2 x  4

1. Complete:
a) 5x  10 y  5x  ......
c) a 2 b  6a 2 b 2  a 2 b......  ......

b) 6 x 2  8x  2 x3x  ......
d) 2ab  5ac  a 2  a2b  .....  .....
2. Extract common factor:
a) 3abc  5bc  7abcd  2bcd 
b) 5x 2 y 3 z 7  3x 8 y 4 z  11x 5 y 3 z 4 
c) 14 x 3 y 2 z 4  7 x 5 y 3 z 7  21x 2 y 5 z 3 
d) 54n 4 m 6  18n 3 m 5  27n 7 m 3 
e) 5 x  10 xy  5 x 2 y 
f) 3x 5 y 4  9 x 2 y 3 3xy  3 y 
g) 5 y 2 x  15 yx 2  y 3 x 4 
h) 6 x 2 y 2  9 x 3 y 6  27 xy3 
i) 9 x 3 y  12 x 2 y 2  18xy3 
j) 4x + 8y =
k) 15x2y3 + 10xy2 =
ACTIVITIES
1. Do the questions carefully. (39 questions)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
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14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
2. Vocabulary
Here are some words used in algebra:
brackets
unknown
equation
value
expression
variable
formula
solution
term
Complete the following statements to show that you understand the meaning of these
words.
1. An unknown is _______________________________________________________.
2. A variable is _________________________________________________________.
3. The difference between an equation and an expression is ______________________
______________________________________________________________________.
4. A formula is _________________________________________________________.
5. In algebra, brackets are _______________________________________________ .
6. Another word for solution is ____________________________________________ .
7. You find the value of the formula when you ________________________________
8. A term is a ___________________________________________________________
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3. Expressions
This is an expression:
We can write this in words as:
n+3
Add three to a number.
Now write the following expressions in words.
1. 5 − n
2. n + 1
3. 2n − 3
4. 6 − 2n
n
5.
3
6. n2
C. Inverse
What’s the meaning of inverse?
Operation
1. addition (+)
2. division (:)
3. multiplication (·)
6. square (x2)
7. square root (
)
8. subtraction (-)
The words on the right are all
operations used in algebra.
Write the inverse operation
alongside each term. The first
one has been done for you.
3. Evaluate 6a + 4b + 5c when:
1) a=1, b=1, c=1
2) a=2, b=-1, c=3
4. Multiply out the brackets and simplify:
a) 3(a+4) =
c) 5(x-2)+4(y-2) =
Inverse operation
_______________
_______________
_______________
_______________
_______________
_______________
3) a=3, b=-3, c=-4
b) 4(a+2)+3(b+1) =
d) 3(2x-3y+z)+2(3x+5y)=
5. Word problems as equations.
When converting word problems to equations, certain "key" words tell you what kind of
operations to use: addition, multiplication, subtraction, and division. The table below
shows some common phrases and the operation to use.
Word
Operation
Example
sum
Addition
difference
subtraction
product
times
less than
multiplication
multiplication
subtraction
total
Addition
more than
Addition
The sum of my age and 10 equals 27.
The difference between my age and my
younger sister's age, who is 11 years old, is
5 years.
The product of my age and 14 is 168.
Three times my age is 60.
Seven less than my age equals 32.
The total of my pocket change and 20
dollars is $22.43.
Eleven more than my age equals 43.
As an equation
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