4.1 Common Factors in Polynomials
Expanding and Factoring are _______________ operations.
(x  2)(x  1)  x2  x  2
Investigate:
Recall: The GCG (greatest common factor) is the greatest number and/or variable that
is a factor of two or more numbers or terms. Hint: factor implies division.
Determine the GCF (greatest common factor) for each of the following:
i) 3, 9, and 12:
ii) 4, 6 and 10:
Now try factoring out the GCF from each example above:
3  3(1)
9  3(3)
12 
ii)
450
400
350
Revenue ($)
i)
300
250
200
150
100
50
-15 -10 -5
5 10 15 20 25 30 35
Price Change ($)
Let’s try with variables:
2
3 5
i) Determine the GCF of x , x , x :
ii) Factor out the GCF from all three terms:
Example 1: Find the greatest common factor (GCF): 3x5 y, 9 x 4 y 6 , 12 x 2 y
The greatest common factor of the coefficients is: _______________
The greatest common factor of the variable parts is: _____________
Therefore, the greatest common factor of the polynomial is: ________
Factor out completely: 3x5y  9x4y6  12x2y
Example 2: Monomial Common Factor
Factor fully.
a) 2x + 6x2
b) 6x2 + 12x
c) 8x3 – 6x2y2 + 4x2y
Example 3: Binomial Common Factor
Factor fully.
a) 3x(2y + 1) + 4x(2y + 1)
b) 8y(3z – 4) – (3z – 4)
Example 4: Factoring by Grouping
Factor fully.
a) ac + bc + ad + bd
b) 2m2 + 6n + 4m + 3mn
Example 5: Factor the following
i) 2x2  6x
i) 2m  3  ym  3
ii) 32x3 y 4  16x5 y2z
ii) 3xx  1  21  x 
iii) 24m3n  36m2n2  42mn 3
iii)
xx  2  32  x 
```````
What is it?
It is a monomial, with the
greatest numerical
coefficient and greatest
degree that is a factor of
all the terms in the
polynomial.
Tips & Tricks
For coefficients:
 take the greatest number
that divides evenly into
the 2 or more
coefficients.
Steps:
1. Find the GCF
For variables:
 take the lowest exponent
of each letter, common to
all terms.
3. Divide each term of the
polynomial by the GCF,
and write the answer
inside the brackets.
2. Write the GCF on the
left of a set of
brackets.
Note: If a letter is NOT
common to all terms, then
that letter does not form
part of the GCF.
Example 6: Differentiated Instruction
a) Write a quadratic trinomial whose greatest common factor is 3x.
Tip: A polynomial is considered completely factored when no more variable factors can
be removed and no more integer factors, other than ______ or _______, can be
removed
Example:
-5x + 3y
ax2  bx  c
4.3 Factoring Simple Trinomial Quadratics:
What is this “a” you speak of?
Trinomials come in the form
ax 2  bx  c
Steps:
1. First remove any GCF, if possible
2. Find two numbers whose product is (a x c) and whose sum is ‘b’.
 “a” is the coefficient in
front of x2.
x2 – 5x – 24
e.g.
___ x ___ = -24
 “b” is the coefficient in
front of x .
 “c” is the third term
(constant term).
Tips & Tricks
The value of “a” has to be 1.
___ + ___ = -5
3. Write 2 sets of brackets.
(
)(
)
4. Write x as the first term in each bracket.
(x
) (x
)
5. The second terms are the two numbers you found in
step 2. Final answer is:
( x - 8) ( x + 3)
Examples
1. Factor.
a) x2 + 7x + 12
b) m2 - 4m - 12
c) a2 – 10a + 21
d) x2 – 4x – 32
2. Factoring by finding the GCF first.
a) 4x2 + 16x - 48
b) 3x2 – 6x – 72
c) 5x2 + 15x – 50
4.4 Factoring Complex Trinomial Quadratics:
ax2  bx  c
Trinomials of the form ax2 + bx + c where a  1, are called complex
trinomials. The rules found to factor simple trinomials cannot be used for
these type of trinomials.
TIP:
When factoring a trinomial, first remove any common factors. Then, factor fully.
Example 1: Factor 3x2 - 11x – 4
Product
of ac
Sum
of b
Steps:
1. ________________________________
2. ________________________________
3. ________________________________
4. ________________________________
5. ________________________________
6. ________________________________
7. ________________________________
Example 2: Factor the following:
a) 2x2 - 5x - 3
Product
of ac
Sum
of b
c) 70 x 2 + 21 x – 7
Product
of ac
b) 6x2 + 7x + 2
Sum
of b
Product
of ac
Sum
of b
Product
of ac
Sum
of b
d) -20 x 2 - 24 x + 32
Example 3:
Over the last season, the manager at Mexx used the quadratic relation
R = 300 + 20x – x2 to model the effect of raising or lowering the price on revenue.
R, is the revenue in dollars and x is the price change in dollars.
a) Confirm that for a $10 price, there will be 40 sales.
b) At what price change does Mexx have no revenue?
c) What is the price change that results in revenue of $375? $300?
GIZMO : http://www.explorelearning.com/index.cfm?method=cResource.dspDetail&ResourceID=110
What do you
mean
a ≠ 1?
ax b x  c
2
Steps: 6x2 + 13x – 5
1. First remove any GCF, if possible.
2. Find two numbers whose product is ‘a x c’ and whose sum is ‘b’.
e.g. 6x2 + 13x – 5
___ x ___ = -30
___ + ___ = 13
3. Replace the middle term by two terms whose coefficients are the two numbers you
found in step 2.
6x2 + 13x – 5
6x2 + 15x – 2x – 5
4. Group the first two terms and the last two terms.
(6x2 + 15x) (– 2x – 5)
5. Factor a GCF from the first group and then factor a GCF from the second group.
3x(2x + 5) –1(2x + 5)
6. Remove a common binomial factor. Final answer is:
(2x + 5) (3x – 1)
Step 5:
Crucial Step!!
Careful!
Examples
Factor.
a) 6x2 + 11x – 10
b) 5n2 – 7n + 2
c) 6a3 – 14a2 + 4a
d) 8x2 – 10xy -3y2
e) 8m2 + 3mn - 5n2
f) 16x2 + 24xy + 9y2
4.5 Factoring Quadratics: Special Cases
Difference of Squares:
a2 - b2 = (a + b)(a – b)
*both terms must be perfect squares
Example 1: Factor the following
a) x2 – 25
b) 1 - 49x2
d) ( x  4)2  9
e) 4x2 – 16
c) 3a 3  12ab 2
Perfect Square Trinomials:
a2 + 2ab + b2 = (a + b)2 or a2 - 2ab + b2 = (a - b)2
In a perfect trinomial, the following conditions are met:
 The first and last term are perfect squares.
 The middle term is twice the product of the square roots of the first and
last terms.
Example 2: Verify that 4 x 2  20 x  25 is a perfect square trinomial
Example 3: Factor the following:
a) x2 + 10x + 25
b) 4x2 + 12x + 9
c) 3x2 + 12x + 12
A difference of what?!
This is a technique of factoring
applied to an expression of the
form a2 – b2, which involves the
subtraction of two perfect
squares.
A perfect square is a number
found by squaring an integer.
Examples of Perfect Squares:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100
Tips & Tricks:
 Only works with
binomials
 It must be a
difference
(subtraction)
between two perfect
squares
 Constants and
coefficients should
be “perfect squares”
a  b  a  b a  b
2
2
Steps:
49m2 – 25n2
e.g.
1. First remove any GCF, if
possible.
2. Write 2 sets of brackets.



3. Square root each term and put
your answer in both brackets
 7m
5n 7n
5n 
4. Write in a + sign in the first
bracket and a – sign in the
second bracket or vice versa.
 7m  5n 7n  5n 
Examples
1. Factor.
a) 121x2 – 64y2
b) 50a2 -2b2
c) 32m2 – 40n2
2. Why is 9n2  4 not a perfect square? How can you recognize a binomial that is a perfect
square?
4.4 Factoring Challenges
Completely factor the following:
1) 8 x 2 ( x  1)  2 x( x  1)  3( x  1)
2.) x 2  4 x  4  9 y 2
4.6 Reasoning about Factoring Polynomials
Example 1:
The trinomial 8x3  6x2  5x represents the volume of a rectangular
prism. Determine the dimensions of the rectangular prism.
Example 2: Factor each expression
a) x2  x  132
b) 16x 2  88x  121
Example 3: Factor each expression
a)  18x 4  32x2
c)
25a2 9b2

64
49
b) x5y  x2y3  x3y3  y5
d) 4(c  5)4  12(c  5)2  9
6.1 Solving Quadratic Equations
Quadratic Equation: an equation that contains at least one term whose
highest degree is 2; for example, x2  x  2  0
Solving a Quadratic Equation: To solve for “x”.
 From Standard Form y  ax2  bx  c  0 , we can factor to y  a(x  r)(x  s) and
solve for x
 In this section, you will learn to solve quadratic equations for any value of “y”.
The graphs below show the 3 possible solutions when solving a quadratic equation
2 zeros
1 zero
no zeros
Solving by Factoring
The zero product property states that if the product of 2 numbers
is zero, then one or both of the numbers must be zero
Example 1:
Solve (x  3)(x  1)  0
Using the zero product property:
To Solve a Quadratic Equation:
1.
2.
3.
4.
Replace the y term with the given value.
Rearrange the quadratic equation to standard form: ax2  bx  c
Factor the quadratic equation, if possible.
Set the factor each to zero and solve for x.
Example 2:
Solve the following quadratic equation y  x2  8x  12 for when y = -3.
Example 3:
Solve the following equations by using the most efficient method:
a) 0  2x2  18
b) y2  30  7y
c) 2x2  9x  4  0
d) 3x(x  6)  50  2x2  3(x  2)
Example 4:
A trendy boutique sells blouses. Over the last season the manager used the quadratic
relations R  x2  20x  300 to model the effect on revenue of raising or lowering the
price. R is the revenue in dollars and x is the price change in dollars. The relation is
graphed below.
1. Use the graph to determine the price change
that produces the maximum revenue.
450
400
2. At what amount of increase or decrease does
the shop get no revenue?
Revenue ($)
350
300
250
200
150
100
50
3. What is the price change that results in a
revenue of $375?
-15 -10 -5
5
10 15 20 25 30 35
Price Change ($)
4. What is the price change that results in a revenue of 300?
Equation 2
Equation 3
 x2  20x  300  0
 x2  20x  300  375
Equation 4
Example 5:
A ball is thrown from the top of a seaside cliff. Its height, h, in meters, above the sea
after t seconds can be modelled by h  5t 2  21t  120 .
a) How high is the cliff? (h-intercept)
b) How long will the ball take to fall 20 m below its initial height?
6.4 Quadratic Formula
Quadratic Formula: formula to help determine the roots of a quadratic
equation
that cannot be factored.
There are three ways to determine the roots:
 Factoring and solving for x
 Quadratic Formula
x
- b  b 2  4ac
2a
Developing the Quadratic formula: (could also be done by completing the square)
Explain each step in the derivation of the quadratic formula.
Deriving the Quadratic Formula
Explanation
ax 2  bx  c  0
4a2x2  4abx  4ac  0
4ax 2  4abx  4ac
4ax 2  4abx  b2  4ac  b2
2ax  b2ax  b  4ac  b2
2ax  b2  b2  4ac
2ax  b   b2  4ac
2ax  b  b2  4ac
x
 b  b 2  4ac
2a
Multiply both sides by 4a
Example 1:
Determine the solutions to each equation using the most efficient method:
a) x2  7x  6  0
Solutions of a quadratic equation in the
form ax2  bx  c =0 can sometimes be
found by FACTORING
b) 3x2  10  8
Quadratic equations that do not contain
an x-term can be solved by isolating the
x-term.
c) 2(x  1)2  16  0
Quadratic equations in vertex from can
be solved by isolating the x-term.
d) x2  5x  2  0
When a quadratic equation CANNOT be
factored, we can solve by using the
Quadratic Formula
Example 2:
Solve 3x(5x  4)  2x  x2  4(x  3) . Round your solutions to 2 decimal places.
Example 3:
A picture frame measures 12 cm by 20 cm. We wish to add a border of equal width
around the frame. The area of the border must equal to the area of the frame.
Determine the width of the border correct to one decimal place.
First you take a negative ‘b’
Do plus or minus, and follow me
Then under a big square root you’ll see
‘b2’ minus ‘4ac’
Divide it all by 2 times ‘a’
That’s how you find x today
6.5 Interpreting Quadratic Equation Roots
Roots: a solution; a number that can substituted for a variable to make the
equation true; for example x = 1 is a root of x 2  x  2  0 , since
12  1  2  0
 The roots/zeros of a function are also known as the x-intercepts
 Recall : The graphs show the 3 possible solutions when solving a quadratic equation
10
10
8
8
6
6
4
4
2
2
-10 -8 -6 -4 -2
2
4
6
8 10
-10 -8 -6 -4 -2
-2
2
4
6
8 10
-2
-4
-4
-6
-6
-8
-8
-10
-10
10
8
6
4
2
-10 -8 -6 -4 -2
2
4
6
8 10
-2
-4
-6
-8
-10
The Discriminant


The portion of the quadratic formula, b2  4ac , is called the Discriminant.
The discriminant is needed to tell us the number of roots (1, 2, or 0) and the
type of roots (real and distinct, equal, or no real roots).
Example 1: Complete the following table.
Relation
y  x2  2x  9
y  2 x 2  12 x  18
y  3x 2  4
Sketch of graph
Number
of zeros
Roots calculated using the quadratic
formula
Quantity under
the square root,
b2 – 4ac
(+, - , or, 0?)
Compare the “ Number of Zeros” column with the column labelled, “Quantity Under the
Square Root.” Under what conditions does a quadratic relation in standard form have:
a) two real roots?
b) two equal real root?
c) no real roots?
Example 2: Without solving, determine the number of real roots of each equation and
describe the graph of the relation.
a) 3x2  4x  5  0
b) -2x2  7x  1  0
c) 9x2 -12x  4  0
SUMMARY:
In the quadratic formula, the value of the discriminant, b2  4ac , determines the
_______________ and _________________ of roots a quadratic equation has.



If b2  4ac > 0, then the quadratic equation has _______________________
If b2  4ac = 0, then the quadratic equation has _______________________
If b2  4ac < 0, then the quadratic equation has ___________________________
6.6 Solving Problems Using Quadratic Models
Strategizing: When solving a problem that involves a quadratic relation, there
are ways to approach which method is most useful:
i)
Vertex Form: to determine the maximum or minimum value of the
relation
ii)
Standard/Factored Form: to determine the value of x that corresponds
to a given y-value of the relation. You may need to use the quadratic
formula. (Zeros/Roots)
Example 1:
The volunteers at a food bank are arranging a concert to raise money. They have to
pay a set fee to the musicians, plus an additional fee to the concert hall for each
person attending the concert. The relation P  n 2  580n  48000 models the profit, P,
in dollars, for the concert, where n is the number of tickets sold.
a) Calculate the number of tickets they must sell to break even.
b) Determine the number of tickets they must sell to maximize profits.
Example 2:
Alexandre was practising his 10 m platform dive. Because of gravity, the relation
between his height, h, in metres, and the time, t, in seconds, after he dives is
quadratic. If Alexandre reached a max height of 11.225 m after 0.5 s, and he hit the
surface of the water at 6.5 seconds, determine an equation to model his dive.
Example 3:
Statisticians use various models to make predictions about population growth.
Ontario’s population (in 100 000s) can be modelled by the relation
P  0.007 x 2  0.19 x  21.5 , where x is the number of years since 1900.
a) Using this model, what was Ontario’s population in 1925?
b) When will Ontario’s population reach 15 million?
10
8
6
4
2
-10 -8 -6 -4 -2
2
-2
-4
-6
-8
-10
4
6
8 10
Example 4:
Lila is creating dog runs for her dog kennel. She can afford 30 m of chain-link fence to
surround four dog runs. The runs will be attached to a wall, as shown in the diagram.
To achieve the maximum area, what dimensions should Lila use for each run and for
the combined enclosure?
WALL