Worksheet 5
Factoring Quadratic and Non-Quadratic Polynomials
Name: ___________________________________________ Section: _______
Basic Vocabulary: Let’s look at the difference between a Polynomial versus a
Non Polynomial:
Polynomials
Not Polynomials
4x 5 y + 7x
5x -3 + 2x
6
- 5x +1
x2
-5x 3 + 2x +
2
3
problem: negative integer exponent
problem: variable in denominator
A Polynomial is a sum of terms (finite) in which all variables are raised to nonnegative powers and no variables appear in the denominator.
A polynomial is a general term. The following are some specific way you can name a
Polynomial:
Monomials (one term)
Binomials (two terms)
Trinomials (three terms)
ax 2
x+y
-3x
6y 2 - 2
-x 4 + 3x 3 +1
4
5 3
z - 2z
7
8y 2 - 2y -10
x 2 + 4xy + y 2
The degree of a polynomial is the largest degree of all terms.
Exercise 1: Find the degree of each term
a) 3x 2
d) 5
deg=2
deg=
b) -23 x 5
e) 12x 2 yz3
deg=
deg=
c) y
deg=
Exercise 2: Find the degree of the polynomial P(x) = 3xy + x 2 y2 - 5x 2 - 6
Note: When adding or subtracting polynomials, you can only combine like terms.
The following table will help you distinguish between like and unlike terms:
Like Terms
Unlike Terms
-5x 2 , -x 2
4x 2 , 3x
7xy3z, -2xzy3
12x 2 y3, -2xy3
Note: If the variable part matches exactly, you can call the terms like terms. To
simplify a polynomial, use the reverse distributive property and combine like terms.
Here is how you can use the distributive property also known as FOILING or
MULTIPLYING POLYNOMIALS:
Example 1: Multiply and simplify the following product
a) 2x(5x-4)
b) 3(x+5)+2(1-3x)
c) 6+4[-2(5-3x)-(5x_3)]
d) (1-x)(1+x)
e) (2-x)(2+x)
f) (x+4)(x+2)
Do you see a pattern emerging from (d) and (e)?
Factoring is the reverse process of multiplying out. It is the process of writing the
polynomial as a product of other factors.
Following are the different methods of Factoring Polynomials:
1) Common Factors: Factoring by Grouping
2) Factoring Trinomials
3) Factoring by using special Factoring Formulas
Method 1: Common Factors
To factor a polynomial by taking out the common factor, we first factor out the
greatest common factor (GCF) of its terms, using the distributive property.
The GCF is a list of terms or monomials consists of the product of:
i)
The greatest numerical factor common to the coefficient in every term
ii)
The highest exponent of the variable factor common in every term
Example 2: Find the GCF of
.
The coefficients (6, 12, 21, and 9) share the greatest common factor of 3.
Only the variables x and z appear in every term.
The GCF is
. The highest power of x is 1 and the highest power of z is 1.
To factor out the GCF, divide each original term by the GCF to obtain the factor.
Exercise 3: Factor the following as completely as possible by factoring out the
common factor:
a) 8x 3 + 20x - 28 =
b) 12a3b -8a2 c2 + 6ab3 =
c) 15x 3 y 4 - 5x 2 y3
d) q 5 p4 - q 4 p3 + qp2
Factoring out by grouping: When we factor out a common factor, we will do it in
two steps. First by grouping them and factoring out a common monomial. Then we
will factor out a common factor (not a monomial) from the grouped terms.
Example 3: Factor 6 xy  8 x  9 y  12 by grouping.
Begin by factoring 2x from the first two terms and 3 from the last two terms.
6 xy  8 x  9 y  12
2 x(3 y  4)  3(3 y  4)
Notice that (3 y  4) is a factor to both terms. Factor it out of each
2 x(3 y  4)  3(3 y  4)
(3 y  4)( 2 x  3)
Exercise 4: Factor the following by grouping.
a) 2 x 2  16 x  xy  8 y
c) x 2  5 x  2 x  10
b) 15u 2  24uv  10u  16v
d) 3 x 2  6 x  2 x  4
Method 2: Factoring Trinomials
Remember: Trinomials are a variety of polynomials that have three terms.
For example: x 2 + 6x +8 is a trinomial. Note, that there is no common factor to
factor out in this case. We are going to learn how to rewrite the trinomial into a
product of two binomials. It is the opposite direction of work from multiplying out
two binomials.
Recall the solution of Example 1 (f): (x+4)(x+2) =
Here is how you can work backwards if you are given x 2 + 6x +8 to factor.
Example 4: Factor: x 2  6 x  8
First, we find the factors of 8 that have a sum of 6. In this case, they are 4
And 2. Then, rewrite the center term using these factors. Lastly, factor by
grouping.
x2  6 x  8
 x 2  4 x  2 x  8 center term is rewritten
 x( x  4)  2( x  4) factor by grouping
 ( x  2)( x  4)
Here is a X method to do the same problem but in a different way.
But suppose you have a trinomial with a leading coefficient different than1. Here is
how you go about factoring it:
Example 5: Factor: 2 x 2  7 x  3
In order to factor this, we look at 2  3  6 and find two factors of 6 that add
Up to 7. The factors that work are 1 and 6. Then rewrite the center term
Using these factors. Lastly factor by grouping.
2 x2  7 x  3
 2 x 2  6 x  1x  3 center term is rewritten
 2 x( x  3)  1( x  3) factor by grouping
 ( x  3)(2 x  1)
Example 6: Factor: 10 x 2  11x  6 . In this case, we need to find the factors of
10  6   60 that add up to give -11. The factors of -60 that works are -15 and 4.
10 x 2  11x  6
 10 x 2  15 x  4 x  6
 5 x(2 x  3)  2(2 x  3)
 (2 x  3)(5 x  2)
Exercise 5: Factor the following.
a) x 2  3x  18
b) x 2  7 x  10
c) x2- 4x + 4
d) 16c 2  24c  9
e) 3 x 2  7 x  10
f) 2 x 2  3 x  9
Method 3: Factoring by using special Factoring Formulas
Simplify the following expressions.
A)
( x  3)( x  3)
B)
(4 x  3)( 4 x  3)
What do you observe? When you multiply a sum and difference of two terms, your
answer is difference of two squares of each of those terms
Here is the Special Product Formula called Difference of Two Squares :
x 2 - a2 = (x + a)(x - a)
Example 7: Factor: 16 x 2  9 . We know that both 16x 2 and 9 are perfect squares,
and that there are only two terms, not three. This is how we know it’s a difference
of squares, and that we are not factoring perfect squares like the above example.
The square root of 16x 2 is 4x , and the square root of 9 is 3.
16x 2 - 9 = (4x)2 - (3)2 = (4 x  3)( 4 x  3)
Example 8: Factor: 3x 2  12 . We know that 3x 2 and 12 are not perfect squares, but
we know that we can factor out a GCF from both terms.
3( x 2  4) = 3éë(x)2 - (2)2 ùû Now we can factor using the difference of squares.
3( x  2)( x  2)
Exercise 6: Completely factor the following:
a) x 2  49
b) 144 x 2  64
c) 2 x 2  32
d) 9y2 - a2b2
Simplify the following expressions by FOILING:
e)
(2 x  5)( 2 x  5)
f)
(3x  2)(3x  2)
What do you observe? This is called a perfect square trinomial. A trinomial is
called a perfect square trinomial if it can be written so that its first term is the
square of some quantity a, its last term is square of some quantity b, and its middle
term is twice the product of the quantities a and b.
Here is the Special Product Formula called a Perfect Square Trinomial:
( x + a) = ( x + a) ( x + a) = x 2 + xa + ax + a2 = x 2 + 2ax + a2
2
( x - a) = ( x - a) ( x - a) = x 2 - ax - xa + a2 = x 2 - 2ax + a2
2
Example 8: Factor 9x 2 + 30x + 25
9x 2 + 30x + 25 = (3x)2 + 2 × 3x × 5+ (5)2 = (3x + 5)2 = (3x + 5)(3x + 5)
x2
+ 2 × x × a + (a)2 = ( x + a) = (x + a)(x + a)
2
Example 9: Factor x 2 -12x + 36
x 2 -12x + 36 = ( x ) - 2 × x × 6 + ( 6) = ( x - 6) = (x - 6)(x - 6)
2
( x)
2
2
2
- 2 × x × a + ( a) = ( x - a) = (x - a)(x - a)
2
2
Exercise 7: Completely factor the following.
a) 9 x 2  6 x  1
b) 100 x 2  60 x  9
c) x 4  4 x 3  4 x 2
d) 4 y 3  28 y 2  49 y
2
e) 100x - 220x +121
Remark: 1) If a polynomial can not be factored, it is called a prime polynomial
2) We don’t have any factorization for the sum of two squares x 2 + y 2
Summary on how to approach factoring a polynomial completely:
Step1: Are there any common factors? If so, factor out the GCF
Step 2: How many terms are in the polynomial?
a) If there are two terms and their squares are subtracted from each other, use
the difference of two squares formula
b) If there are two terms and their cubes are added or subtracted from each
other there is another special formula that you will get to learn later.
c) If there are three terms first check if it is a perfect square trinomial and use
2
2
the appropriate special formula ( x + a) = x 2 + 2ax + a2 ,
( x - a) = x 2 - 2ax + a2
d) If there are three terms and it does not fit into the special formula then factor
by using the X method.
e) If there are four or more terms, try factoring by grouping.
Step 3: See if any factors in the factored polynomial can be factored further.
Exercise 8: Factor completely. If a polynomial cannot be factored, state that it is
prime.
a) 3x 2  9 x  12
b) x 3  64 x
c) x 2  11x  28
d) 25 x 2  49
e) 3 x 2  75
f) 3x2 - x - 4
g) xy 2  9 x  5 y 2  45
h)  10 x 4 y 2  5x 3 y 3  15x 3 y 2
Application
The volume of a rectangular box x inches in height is given by the relationship
V = x 3 + 8x 2 +15x. Factor the right hand side to determine
a) The number of inches the width exceeds the height by
b) The number of inches the length exceeds the height by
c) The volume of the box given the height is 3 ft.