Unit 7 Factoring Polynomials
Introduction
In Unit 2 we learned how to solve linear equations. Linear equations have a
single variable whose exponent is one. Below is an example of a linear equation.
3(x + 2) – x + 4 = 4x – 22
In this equation the only variable we use is ‘x’ so this is the single variable from our
definition above. Each instance of ‘x’ could actually have the understood exponent of
one written with it, if we needed an exponent. If we added the understood exponent
of one, our example equation would be:
3(x1 + 2) – x1 + 4 = 4 x1 – 22
The above equation has one variable ‘x’ and in each case there is an exponent of
one. These equations are very useful. The slope-intercept form of a linear equation
is y = mx + b. We saw many ways in Unit 4 in which this type of equation can model
situations that have rates of change in Unit 4. A linear equation can model a company’s
sale, a person’s pay, the altitude of a balloon, the relationship or a person’s height
to foot size, etc.
Linear equations are not the only type of equation that we can use to model
situations. In this unit we are going to learn about a type of equation called
quadratics. These equations have a squared term. An example of a quadratic equation
is given below.
x2 + 5x = -6
We will need to learn how to solve this type of equation. One of the primary techniques
that can be used to solve a quadratic equation is called factoring. Factoring is a term we should know. Factoring means we should find numbers that are multiplied together. In this unit we are going to find expressions that can be multiplied together.
Unit 7 Vocabulary and Concepts
Factor Used as a verb, it means to find the values of monomials or binomials
that multiply to produce a given polynomial.
Key Concepts for Exponents and Polynomials
Raising a Monomial to a Power To raise a monomial to a power, we multiply the
power times ALL the exponents in the monomial.
Types of pairs of factors Pairs of factors can be positive, negative, or mixed.
Standard Form for a Polynomial Standard form for a polynomial has a first term
that is positive, and all of the terms are arranged
in descending order of exponents.
Zero Product Theorem If two numbers have a product of zero then one, or both,
of the values must be zero. If ab = 0 then a = 0 or b = 0.
Trial and Error Factoring Used when the highest ordered term has a coefficient other
than one. We must try to use the factors of the first
and last terms to produce the middle term.
Difference of Perfect Squares A type of binomial factored into sum and difference
binomials. Two perfect squares are separated by a
minus sign. Ex: a2 – b2 = (a + b)(a – b)
Objective
· The student will raise a monomial to a power.
A number or variable raised to a second power means to multiply two of the variables
or number times each other. A number or variable raised to a third power means to
write down the factor three times and multiply them together. A number or variable
raised to a fourth power means to write down the factor four times and multiply them together. A monomial raised to the ‘n’th power means to write down the monomial
‘n’ times and multiply them together. The examples below show us how to raise a
monomial to a power.
Example A Example B Example C
(5x3)2 (4x5y)3 (-2a3)4
means means means
(5x3)(5x3) (4x5y)(4x5y)(4x5y) (-2a3)(-2a3)(-2a3)(-2a3)
and results in and results in and results in
25x6 64x15y3 16a12
Writing out the monomials the required number of times and multiplying will produce
correct answers every time. However, when the power becomes larger and larger,
writing out the monomial six or seven times would be inefficient. We need to examine
these examples closely so we can draw some conclusions. First, we should try to find a
pattern in the way the exponents work. The table below may help us understand the
process better.
Exponent Power Resulting
inside the outside the Exponent
parentheses parentheses
Example A 3 2 6
Example B 5 3 15
Example B 1 3 3 ( the understood exponent of ‘y’ )
Example C 3 4 12
If we ask ourselves, “What arithmetic do we use with the exponent from the monomial and the
power to get the resulting exponent?”, the answer appears to be to multiply! And this gives us
our rule for “raising a monomial to a power”.
This rule applies to ALL exponents in the monomial. This includes the understood exponents of both the variables and the coefficients. If we examine the coefficients
in examples A through C on the previous page we can see this clearly.
Coefficient Power Result
Example A 51 2 52 = 25
Example B 41 3 43 = 64
Example C (-2)1 4 (-2)4 = 16
Our approach to raising a monomial to a power should be to put in ALL the understood
exponents, then multiply all the exponents by the power, and simplify the coefficient.
The examples below show us how to use the process.
Example A Simplify (3a2b)4
(3a2b)4
(31a2b1)4 Put in the understood exponents.
34a8b4 Multiply the power ‘4’ times each of the monomial’s exponents.
81a8b4 Simplify the coefficient.
Example B Simplify (-2x3yz-2)5
(-2x3yz-2)5
(-21x3y1z-2)5 Put in the understood exponents.
(-2)5x15y5z-10 Multiply the power ‘5’ times each of the monomial’s exponents.
-32x15y5z-10 Simplify the coefficient.
In this example we needed to be careful of the negative coefficient and the
negative exponent.
Example C Simplify (-3n2m5)4
(-3n2m5)4
(-31n2m5)4 Put in the understood exponent.
(-3)4n8m20 Multiply the power ‘4’ times each of the monomial’s exponents.
81n8m20 Simplify the coefficient.
In this example we had a negative coefficient, but because it was raised to an
even power the simplified coefficient is a positive 81.
Example D Simplify (6x4yz-3 )-2
(6x4yz-3 )-2
(61x4y1z-3 )-2 Put in the understood exponents.
6-2x-8y-2z6 Multiply the power ‘-2’ times each of the monomial’s exponents.
x-8y-2z6 Simplify the coefficient.
The negative power must be used carefully. In particular, we must remember
a negative exponent means the value or variable belongs to the denominator of
a fraction. That is why we have a fractional coefficient.
Example E Simplify (3a5bc)2(-2ab2)3
This example shows us using monomials raised to a power, and multiplication.
To simplify this expression we must use the Order of Operations and use the
powers first, and then do the multiplication.
(3a5bc)2(-2ab2)3
(9a10b2c2)(-8a3b6) We raise each monomial to their powers first.
-72 a13b5c8 Now we multiply the coefficients, and add the exponents.
VIDEO LINK: Youtube: Power to a Power
Exercises Unit 7 Section 1
Fill in the blanks.
1. If an expression has a monomial that is raised to a power then we should put in
the ______exponents, ______the power times
all the exponents and simplify the ______.
Simplify the following
2. (a2)4 3. (x7y3)2 4. (3x2)4 5. (-4a6)2
6. (-x2y)3 7. (0.4x4)3 8. (-5a2)4 9. (-3x2y5)3
10. (.2x5)2 11. (a5)3 12. (-xy)2 13. (1.5y9)2
14. (2xy3)6 15. (a-3)4 16. (w-2)-5 17. (7xy-4)-2
Simplify, then multiply the resulting monomials. Show your work.
18. (ab2)3(a3b)4 19. (xy3)2(x2)4 20. (2a4b3)3(3a5b)2
21. (-ab)3(-a2b)6 22. (x2)3(x3)4(x5)2 23. (4ab-2)2(.5ab3)2
24. Which expression is the completely simplified form of (-2ab2c-1d-4)3
a. b. c. d.
25. Which expression is the completely simplified form of (x2y-3)-2
a. b. c. d.
26. Given the expression (-5w3x-1yz-4)-2
a. Simplify the expression completely.
b. Explain how the coefficient will be raised to the "negative two" power.
c. Explain how you raised w3 to the negative two power and why it
belongs in either the numerator or denominator of your answer.
Read the problems carefully to find the two data points described in the problem, and
then write an equation to model the situation in slope-intercept form.
27. A hot air balloon has lifted off the ground. After 4 minutes, it was 220 feet high.
The balloon is rising at a steady rate. Write an equation to model how the altitude
is increasing. (Hint: What was the balloon’s altitude when it first started, and what
was its time when it started? )
( , ) ( , ) m = ______equation ______
28. A concert at a large arena just finished. After 5 minutes there were 11800 people
still in the arena. After 11 minutes there were 5200 people left in the arena.
a. Write an equation to model how fast people are leaving.
( , ) ( , ) m = ______equation ______
b. How many people were at the concert?
Explain how you found your answer!
c. How many minutes will it take to empty the arena?
Explain how you found your answer!
Objective
· The student will factor trinomials of the form x2 + bx + c where
b > 0 and c > 0.
Solving problems often means reversing operations. If we buy 5 tickets to a show
at $9 each we can use multiplication to find the total. However, if we know we bought
5 tickets for $45 then to find the price of each ticket we must divide. After we learned
how to solve problems with multiplication we then learned how to solve problems with
the reverse, with division.
We can solve area problems like the one below with binomial multiplication.
Given the rectangle to the right with the length and width
listed as expressions, find the expression for the area. 2x + 5
(2x + 5)(x – 1)
2x2 – 2x + 5x - 5 x - 1
2x2 + 3x – 5
If there are problems we can solve with binomial multiplication there will also be problems
we can solve by reversing the process of multiplication.
In reversing the process of binomial multiplication there is a key type of questions we
need to use. A sample of that question follows.
“What two numbers add up to 11 and multiply to 24?”
Finding the answer to this question involves being able to list the pairs of factors for 24.
Below is a list of these pairs.
24
1, 24
2, 12
3, 8 This is the answer since 3 + 8 = 11
4, 6
In listing the pairs of factors we started with the number 1 and worked our way up to
4 in order. By listing the pairs in order we make sure we don’t skip any pairs.
This question and our ability to find the pairs of factors is critical because in the process
of binomial multiplication we will end up multiplying two numbers AND adding those same
two numbers. The examples that follow demonstrate this arithmetic.
Example A Example B Example C
(a + 4)(a + 9) (x + 7)(x + 2) (z + 8)(z + 3)
a2 + 9a + 4a + 36 x2 + 2x + 7x + 14 z2 + 3z + 8z + 24
a2 + 13a + 36 x2 + 9x + 14 z2 + 11z + 24
To reverse the binomial multiplication we must find the two numbers that multiply
to the last term, and add to become the coefficient of the middle term. Below are some examples that show us how to use our question from the previous page to find the
binomial factors when we are given the trinomial product.
Example A Factor x2 + 11x + 30
First we must find “two numbers that multiply to 30 and add up to 11”.
To do this we will list all the pairs of factors of 30.
30
1, 30
2, 15
3, 10
5, 6 This is the pair of factors that adds up to 11.
These are the values that belong in our binomials so our answer is
(x + 5)(x + 6)
You can check your answer by using the FOIL process and multiplying.
Example B Factor a2 + 6a + 9
First we must find “two numbers that multiply to 9 and add up to 6”.
To do this we will list all the pairs of factors of 9.
9
1, 9
3, 3 This is the pair of factors that adds up to 6.
These are the values that belong in our binomials so our answer is
(a + 3)(a + 3), which in this case is best written as (a + 3)2
Example C Factor z2 + 43x + 42
First we must find “two numbers that multiply to 42 and add up to 43”.
To do this we will list all the pairs of factors of 42.
42
1, 42 This is the pair of factors that adds up to 43.
2, 21
3, 14
6, 7
These are the values that belong in our binomials so our answer is
(z + 1)(z + 42)
When we are listing our pairs of factors, if we can see that a pair adds
up to the numbers we need then it may not be necessary to write out
all the pairs. For instance, in the above example the very first pair we
listed was correct so we would not necessarily have to list the rest.
When we list our binomial answers the order of the binomials is not critical since
multiplication is commutative. For instance:
(x + 7)(x + 11) = (x + 11)(x + 7)
The binomials can be written in either order and are correct.
VIDEO LINK: Youtub: Factoring a Simple Trinomail
Exercises Unit 7 Section 2
1. Find a pair of factors that multiply to 20 and add up to 9.
2. Find a pair of factors that multiply to 18 and add up to 19.