7.2: Factoring Polynomials
Introduction
In this lesson you will learn to factor polynomials by removing the greatest common factor. You will also learn to factor polynomials of the form where . You will learn a method known as the ‘box’ method to do this factoring. You will use the same method to factor polynomials of the form where . The last polynomials that you will earn to factor are those known as the sum and difference of two squares.
Objectives
The lesson objectives for Factoring Polynomials are:
- Factoring a common factor
- Factoring where
- Factoring where
- Factoring the sum and difference of two squares
Factoring a Common Factor
Introduction
In this concept you will learn how to factor polynomials by looking for a common factor. You have used common factors before in earlier courses. Common factors are numbers (numerical coefficients) or letters (literal coefficients) that are a factor in all parts of the polynomials. In earlier courses of mathematics you would have studied common factors of two numbers. Say you had the numbers 25 and 35. Two of the factors of 25 are . Two of the factors of 35 are . Therefore a common factor of 25 and 35 would be .
In this concept of Lesson Factoring Polynomials, you will be using the notion of common factors to factor polynomials. Sometimes simple polynomials can be factored by looking for a common factor among the terms in the polynomial. Often, for polynomials, this is referred to as the Greatest Common Factor. The Greatest Common Factor (or GCF) is the largest monomial that is a factor of (or divides into evenly) each of the terms of the polynomial. This concept will be the first explored in Lesson Factoring Polynomials.
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Khan Academy Factoring and the Distributive Property
Guidance
Factor the following polynomial: .
Step 1: Identify the GCF of the polynomial.
If you look at just the factors of the numbers you can see the following:
Looking at the factors for each of the numbers, you can see that 12, 6, and 3 can all be divided by 3.
Also notice that each of the terms has an in common.
So the GCF for this polynomial is
Step 2: Divide out the GCF from each term of the polynomial.
Example A
Factor the following binomial:
Step 1: Identify the GCF.
Looking at the factors for each of the numbers, you can see that 5 and 15 can both be divided by 5.
So the GCF for this binomial is 5.
Step 2: Divide out the GCF from each term of the binomial.
Example B
Factor the following polynomial:
Step 1: Identify the GCF.
Looking at the factors for each of the numbers, you can see that 4, 8, and 2 can all be divided by 2.
So the GCF for this polynomial is 2.
Step 2: Divide out the GCF from each term of the polynomial.
Example C
Factor the following polynomial:
Step 1: Identify the GCF.
Looking at the factors for each of the numbers, you can see that 3, 9, and 6 can all be divided by 3.
Also notice that each of the terms has an in common.
So the GCF for this polynomial is .
Step 2: Divide out the GCF from each term of the polynomial.
Vocabulary
- Common Factor
- Common factors are numbers (numerical coefficients) or letters (literal coefficients) that are factors in all terms of the polynomials.
- Greatest Common Factor
- The Greatest Common Factor (or GCF) is the largest monomial that is a factor of (or divides into evenly) each of the terms of the polynomial.
Guided Practice
- Find the common factors of the following:
- Factor the following polynomial:
- Factor the following polynomial:
Answers
1.
Step 1: Identify the GCF.
This problem is a little different in that if you look at the expression you notice that is common in both terms. Therefore is the common factor.
So the GCF for this expression is .
Step 2: Divide out the GCF from each term of the expression.
2.
Step 1: Identify the GCF.
Looking at the factors for each of the numbers, you can see that 5, 15, 10, and 25 can all be divided by 5.
Also notice that each of the terms has an in common.
So the GCF for this polynomial is .
Step 2: Divide out the GCF out of each term of the polynomial.
3.
Step 1: Identify the GCF.
Looking at the factors for each of the numbers, you can see that 27, 18, and 9 can all be divided by 9.
Also notice that each of the terms has an in common.
So the GCF for this polynomial is .
Step 2: Divide out the GCF out of each term of the polynomial
Summary
Remember that when you factor expressions such as polynomials by taking out a common factor, the first step is to find the greatest common factor for all of the terms in the polynomial. Once you have found this common factor, you must divide each of the terms of the polynomial by this common factor.
The common factor can be a number, a variable, or a combination of both. This means that you need to look at both the numbers in the terms of the polynomial and the variables of the polynomial.
Problem Set
Find the common factors of the following:
Factor the following polynomial:
Factor the following polynomial:
Factoring ax² + bx + c where a = 1
Introduction
When there are no common factors, you have to have an alternate method to factor polynomials. In this concept you will begin to factor polynomials that are known as quadratic expressions. A quadratic expression is one in which one variable will have an exponent of two and all other variables will have an exponent of one. The general form of a quadratic expression is where ‘’ and ‘’ are the coefficients of and , respectively, and ‘’ is a constant. The term is the one that is necessary to have a quadratic expression. These expressions are often the result of multiplying polynomials. In this concept, you will be working with quadratic expressions where the coefficient for ‘’ is equal to 1.
Quadratic expressions of the form can be factored using algebra tiles. To do this, the tiles modeling the trinomial are laid out. Next, with the tile in the upper left position the remaining tiles are arranged around it to form a rectangle. By forming a rectangle, you can easily see the factors of the quadratic expression. Remember you have used algebra tiles before in previous lessons and chapters. The large square tile represents an , the rectangular tile represents the tile, and the small squares represent the 1's. These are shown for you below.
Remember as well that the reverse of these colored tiles are white and this represents the negatives.
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Khan Academy Factoring trinomials with a leading 1 coefficient
Guidance
An electronic sign is used to advertise specials outside a local store. It is shaped like a rectangle. If the dimensions of the sign are known to be represented by the trinomial , find the binomials that represent the length and width of the electronic sign.
The first step to solving this problem is to lay out the tiles you need to represent the trinomial (or the quadratic expression).
The second step would be to arrange these tiles to form a rectangle.
The width of the rectangle is .
The length of the rectangle is .
Therefore the factors of are or .
Example A
Factor
Step 1: Place the tiles that represent the trinomial in front of you.
1 : tile
5 : tiles
6 : 1 tiles
Step 2: Now, form the rectangular using the tiles.
The width of the rectangle is .
The length of the rectangle is .
Therefore the factors of are or .
Example B
Factor
Step 1: Place the tiles that represent the trinomial in front of you.
1 : tile
3 : tiles
2 : 1 tiles
Step 2: Now, form the rectangular using the tiles.
The width if the rectangle is .
The length of the rectangle is .
Therefore the factors of are or .
Example C
Factor
Step 1: Place the tiles that represent the trinomial in front of you.
1 : tile
7 : tiles
12 : 1 tiles
Step 2: Now, form the rectangular using the tiles.
The width if the rectangle is .
The length of the rectangle is .
Therefore the factors of are or .
Vocabulary
- Quadratic Expression
- A quadratic expression is a trinomial in which one variable will have an exponent of two and all other variables will have an exponent of one. The general form of a quadratic expression is where ‘’ and ‘’ are the coefficients of and , respectively, and ‘’ is a constant.
Guided Practice
- Factor .
- Factor .
- Find the binomial dimensions of a rectangle that is represented by the trinomial .
Answers
1. Factor .
Step 1: Place the tiles that represent the trinomial in front of you.
1 : tile
6 : tiles
8 : 1 tiles
Step 2: Now, form the rectangular using the tiles.
The width if the rectangle is .
The length of the rectangle is .
Therefore the factors of . are or .
2. Factor .
Step 1: Place the tiles that represent the trinomial in front of you. In this polynomial you need to find the factors of 8. You can’t build a rectangle with just 2 "" tiles and 8 "–1" tiles.
The factors of 8 are , and . Notice that if you use 4 "" tiles and 2 tiles you will end up with 2 "" tiles left over. Try to build a rectangle with these.
1 : tile
4 : tiles
2 : tiles
8 : tiles
Step 2: Now, form the rectangular using the tiles.
The width if the rectangle is
The length of the rectangle is
Therefore the factors of are or
3. Find the binomial dimensions of a rectangle that is represented by the trinomial .
The first step to solving this problem is to lay out the tiles you need to represent the trinomial.
The second step would be to arrange these tiles to form a rectangle.
The width of the rectangle is .
The length of the rectangle is .
Therefore the factors of are or .
Summary
In this concept you again used algebra tiles but you used them to factor trinomials known as quadratic expressions. These special trinomials were in the form of where . Algebra tiles are useful for factoring quadratic expressions because when you lay out the tiles you need and then form your rectangle, you have the binomial factors of the quadratic. Sometimes this approach is not useful, like when the constant is very large. In these cases, algebra tiles become very cumbersome to use. In the next few concepts, you will begin to explore other methods for factoring quadratics.
Problem Set
Factor the following trinomials.
Factor the following quadratic expressions.
Factor the following quadratic expressions.
Factoring ax² + bx + c where a ≠ 1
Introduction
In the last concept you learned how to factor quadratics or trinomials where . You learned that algebra tiles were very useful as a tool for solving these problems and that by building rectangles, finding the factors of quadratics was a fast way to find the solution. In this lesson, you will work with quadratics, or trinomials, where the value of a does not equal 1. In these problems, algebra tiles may still be useful but may, at some point, become cumbersome due to the large numbers of tiles necessary to build the rectangles. You will use algebra to solve these problems. There are a few other methods that are useful for solving quadratics where . One method you learned earlier is the FOIL pattern. Remember that with FOIL, you multiply the First two terms, the Outside terms, the Inside terms and the Last terms. So for example, with the trinomial:
When you factor trinomials, you are really just using in reverse. Let’s look at the example below. With the trinomial above , you had to factor both the 2 and the –35 (both the first number and the last number). You can say then, in general terms, that with the trinomial , you have to factor both “” and “”.
The middle term is the sum of the outside two and the inside two terms. Therefore
In this lesson you will have the opportunity to work through a number of examples to develop mastery at factoring trinomials of the form where using this reverse FOIL method. Now let's begin.
Guidance
Jack wants to construct a border around his garden. The garden measures 5 yards by 18 yards. He has enough stone to build a border with a total area of 39 square yards. The border will be twice as wide on the shorter end. What are the dimensions of the border?
Remember:
To factor this trinomial try this method:
In this trinomial, the ‘a’ value is 2 and the ‘c’ value is –39. These values are placed in a box
As shown in the illustration below
The product of 2 and –39 is –78. Find the factors for –78 and look for the factor pair that will combine (add or subtract) to give the ‘’ value of +23.
From the list, –3 and 26 will work. Put the factors in the box as shown.
Remember the Greatest common factor from the first part of this Lesson. Going across the horizontal rows in the box, find the GCF of 2 and 26 it will be 2. Find the GCF of –3 and –39 it will be –3. Put these numbers in the box as shown.
THEN - going down the vertical rows...
Find the GCF of 2 and –3 it will be 1. Find the GCF of 26 and –39 it will be 13.
Put these numbers in the box as shown.
These new numbers represent your factors.
So . To find the dimensions of the border:
Since cannot be negative, must equal .
Back to the question!
Width of Border:
Length of Border:
Example A
Factor:
In this trinomial, the ‘’ value is 2 and the ‘’ value is 15. Place these values in a box.
The product of 2 and 15 is 30. Next find the factors for 30 then look for the factor pair that will combine (add or subtract) to give the ‘’ value of +11.
From the list, 5 and 6 will work. Put the factors in the box as shown.
Going across the horizontal rows in the box, find the GCF of 2 and 6. It will be 2. Find the GCF of 5 and 15. It will be 5. Put these numbers in the box as shown.
THEN - going down the vertical rows...
Find the GCF of 2 and 5. It will be 1. Find the GCF of 6 and 15. It will be 3.
Put these numbers in the box as shown.
These new numbers represent your factors.
So
Example B
Factor:
In this trinomial, the ‘’ value is 3 and the ‘’ value is –3. Place the values in a box.
The product of 3 and –3 is –9. Find the factors for –9 and look for the factor pair that will combine (add or subtract) to give the ‘’ value of –8.
From the list, 1 and –9 will work. Put the factors in the box as shown.
Going across the horizontal rows in the box, find the GCF of 3 and 1. It will be 1. Find the GCF of –9 and –3. It will be –3. Put these numbers in the box as shown.
THEN - going down the vertical rows...
Find the GCF of 3 and –9. It will be 3. Find the GCF of 1 and -3. It will be 1.
Put these numbers in the box as shown.
These new numbers represent your factors.
So
Example C
Factor:
In this trinomial, the ‘’ value is 5 and the ‘’ value is 18. Place the values in a box.
The product of 5 and 18 is 90. Find the factors for 90 and look for the factor pair that will combine (add or subtract) to give the ‘’ value of –21.
From the list, –6 and –15 will work. Put the factors in the box as shown.
Going across the horizontal rows in the box, find the GCF of 5 and –6. It will be 1. Find the GCF of –15 and 18. It will be 3. Put these numbers in the box as shown.
THEN - going down the vertical rows...
Find the GCF of 5 and –15. It will be 5. Find the GCF of –6 and 18. It will be 6.
Put these numbers in the box as shown.
These new numbers represent your factors.
So
Vocabulary
- Greatest Common Factor
- The Greatest Common Factor (or GCF) is the largest monomial that is a factor of (or divides into evenly) each of the terms of the polynomial.
- Quadratic Expression
- A quadratic expression is a trinomial in which one variable will have an exponent of two and all other variables will have an exponent of one. The general form of a quadratic expression is where ‘’ and ‘’ are the coefficients of and , respectively, and ‘’ is a constant.
Guided Practice
- Factor the following trinomial:
- Factor the following trinomial:
- Factor the following trinomial:
Answers
Factor...
1.
In this trinomial, the ‘’ value is 8 and the ‘’ value is –3. Place the values in a box.
The product of 8 and –3 is –24. Next, find the factors for –24 and look for the factor pair that will combine (add or subtract) to give the ‘’ value of –2.
From the list, 4 and –6 will work. Put the factors in the box as shown.
Going across the horizontal rows in the box, find the GCF of 8 and 4. It will be 4. Find the GCF of –6 and –3. It will be –3. Put these numbers in the box as shown.
THEN - going down the vertical rows...
Find the GCF of 8 and –6. It will be 2. Find the GCF of 4 and –3. It will be 1.
Put these numbers in the box as shown.
These new numbers represent your factors.
So
2.
First you can factor out the 3 from the polynomial
In this simpler trinomial, the ‘’ value is 1 and the ‘’ value is –20. Place the values in a box.
The product of 1 and –20 is –20. Next, find the factors for –20 and look for the factor pair that will combine (add or subtract) to give the ‘’ value of 1.
From the list, –4 and 5 will work. Put the factors in the box as shown.
Going across the horizontal rows in the box, find the GCF of 1 and –4. It will be 1. Find the GCF of 5 and –20. It will be 5. Put these numbers in the box as shown.
THEN - going down the vertical rows...
Find the GCF of 1 and 5. It will be 1. Find the GCF of –4 and –20. It will be –4.
Put these numbers in the box as shown.
These new numbers represent your factors.
So
3.
First you can factor out the 5 and one from the polynomial
First, use the box for the simpler trinomial to factor it:
The product of 1 and 8 is 8.
Need: factors for 8 and a pair that will combine to give the ‘’ value of 6.
Next: GCFs
Last: Put it together
These new numbers represent your factors.
So
Summary
In the first lesson of this section you learned how to take a common factor out of a polynomial. In the next section you learned to factor trinomials (or quadratics) of the form where using algebra tiles. In this lesson, you continued working with trinomials (or quadratics) but in these quadratics . As well in this lesson, you were introduced to a neat method known as the box method. With this method, any quadratic can be solved by finding the factors that multiply together to give “” and combine to give “”. Using the box method and by finding GCFs, the factors of the quadratic can be found.
Problem Set
Factor the following trinomials
Factor the following trinomials
Factor the following trinomials
Factoring the Sum and Difference of Two Squares
Introduction
In lesson Addition and Subtraction of Polynomials you learned about three special cases where binomials can be expanded using the distributive property to make polynomials. These special cases were
In this lesson, you will again be working with these special cases. The first two special cases are the sum of two squares. These are often called perfect square trinomials. The third special case is called the difference of two squares. Rather than expanding the binomials to make polynomials as you did in the previous lesson, here you will be factoring the special case polynomials. Factoring is really the reverse of multiplication!
You can use algebra tiles or the box method. Recall that algebra tiles are very visual but can be cumbersome when you need to use a lot of tiles to find the factors.
Watch This
Khan Academy Factoring the Sum and Difference of Squares
Guidance
A box is to be designed for packaging with a side length represented by the quadratic . If this is the most economical box, what are the dimensions?
First: factor the quadratic to find the value for .
First, the box:
The product of 3 and –64 is –576.
Need: factors for –576 and a pair that will combine to give the ‘’ value of 0. Start with taking the square root of 576 since you need to have two numbers that are the same. This way they will add to give you 0 and multiply to give 576.
Next: GCFs
Last: Put it together
These new numbers represent your factors.
So
The most economical box is a cube. Therefore the dimensions are .
Example A
Factor
First, the box:
The product of 2 and 98 is 196.
Need: factors for 196 and a pair that will combine to give the ‘’ value of 28. Start with taking the square root of 196 since if this is a perfect square it will require two numbers that are the same. This way the middle terms would be the same and would still multiply to give 196. You could also find the factors of 196. This would give you two numbers that add up to 28 and multiply to give you 196.
Next: GCFs
Last: Put it together
These new numbers represent your factors.
So Or
SPECIAL CASE 1
Example B
Factor:
First, let's factor out a 2 from this expression (the GCF!)
Next the box:
The product of 4 and 9 is 36.
Need: factors for 36 and a pair that will combine to give the ‘’ value of –12. Start with taking the square root of 36 since if this is a perfect square it will require two numbers that are the same. This way the middle terms would be the same and would still multiply to give 36. You could also find the factors of 36. This would give you two numbers that add up to –12 and multiply to give you 36.
Next: GCFs
Last: Put it together
These new numbers represent your factors.
So
Or
SPECIAL CASE 2
Example C
Factor:
First the box:
The product of 1 and 16 is 16.
Need: factors for 16 and a pair that will combine to give the ‘’ value of 0. Start with taking the square root of 16 since if this is a difference of two squares it will require two numbers that are the same. This way the middle terms would be equal and cancel each other out; plus the numbers would still multiply to give 16.
Next: GCFs
Last: Put it together
These new numbers represent your factors.
So SPECIAL CASE 3
Vocabulary
- Difference of Two Squares
- The difference of two squares is a pattern found in polynomial expressions. It is a special case where there is no middle term found in the quadratic expression. The general equation for the difference of two squares is:
- Sum of Two Squares
- The sum of two squares involves two special patterns found in polynomial expressions. These are special cases where the middle term is twice the product of the first and last term. For the quadratic expression , the sum of two squares has a middle term equal to . These trinomials are often called perfect square trinomials. The general equations for the sum of two square patterns are:
Guided Practice
- Factor completely
- Factor completely
- Factor completely
Answers
1.
First the box:
The product of 1 and 81 is 81.
Need: factors for 81 and a pair that will combine to give the ‘’ value of –18. Start with taking the square root of 81 since if this is a perfect square it will require two numbers that are the same. This way the middle terms would be the same and would still multiply to give 81. You could also find the factors of 81.
Next: GCFs
Last: Put it together
These new numbers represent your factors.
So
Or
SPECIAL CASE 2
2.
First: let's factor out a 2 from this expression (the GCF!)
Next the box:
The product of 25 and 49 is 1225. WOW!
Need: factors for 1225 and a pair that will combine to give the ‘’ value of 0. Start with taking the square root of 1225 since if this is a difference of two squares it will require two numbers that are the same. This way the middle terms would equal each other and cancel each other out; plus the numbers would still multiply to give 16.
Next: GCFs
Last: Put it together
These new numbers represent your factors.
So SPECIAL CASE 3
3.
First, let's factor out a 4 from this expression (the GCF!)
Next the box:
The product of 1 and 36 is 36.
Need: factors for 36 and a pair that will combine to give the ‘’ value of 12. Start with taking the square root of 36 since if this is a perfect square it will require two numbers that are the same. This way the middle terms would be the same and would still multiply to give 36.
Next: GCFs
Last: Put it together
These new numbers represent your factors.
So
Or
SPECIAL CASE 1
Summary
In this final lesson you have had the opportunity to apply your skills of factoring to the special cases that you learned about in the previous lesson. Remember the three special cases for polynomials:
A: The Sum of Two Squares
1.
2.
B: The Difference of Two Squares
3.
It is easier if you are able to recognize the special cases in that you do not need to find all of the factors for and . Rather, you simply need to find the square root of the “” term. In special cases 1 and 2, the middle term is always 2. For special case 3, there is no middle term. Both algebra tiles and the box method provide you with useful methods for factoring these special cases as they did with other polynomials.
Problem Set
Factor the following: (Special Case 1 and 2)
Factor the following: (Special Case 3)
Factor the following and identify the case:
Summary
In this lesson, you have worked with adding and subtracting polynomials. You also worked with multiplying polynomials. Remember that when multiplying polynomials, the distributive property is the tool to use. You began your look at the special cases of polynomials and learned that the more you recognize these special cases, the quicker it is to work with them when you are factoring.
In this current lesson, you continued your learning of polynomials by working with the greatest common factor. Make the question simpler by first removing the number or letter that is common in all terms of your polynomial. You also began factoring quadratics (or trinomials) where the coefficient for in was equal to 1 and then for cases where a was not equal to 1. Quadratics where was not equal to one seemed a bit more complex but remember the strategy for solving these remains the same.
Lastly, in this current lesson, you worked with factoring polynomials for three special cases. You may have noticed that these special cases were those you worked with in lesson Addition and Subtraction of Polynomials, only here you were factoring them. Again, strategies for factoring the special case polynomials remain the same. As well, if you are able to recognize a quadratic (trinomial) as a special case, factoring becomes somewhat less cumbersome.