Download Factoring Trinomials - the Algebra Class E

Unit 9: Factoring
Lesson 4: Factoring Trinomials: A Special Situation
Using two different factoring methods.
In the previous lesson, we factored trinomials that had a lead coefficient of 1. Of course,
many trinomials have a lead coefficient that is greater than 1. We will explore one method of
factoring these types of trinomials.
Example 1
Factor: 3x2 – 3x – 36
Notice how the lead coefficient is not 1. However, each term is divisible by 3.
Step 1:
Note: If a trinomial looks difficult to factor, first check to see if a
Greatest Common Factor (GCF) can be factored out.
Example 2
Factor: 4x3 + 4x2 – 24x
Copyright © 2009-2010 Karin Hutchinson – Algebra-class.com
Unit 9: Factoring
Lesson 4: Factoring Trinomials (A Special Situation) Practice
Part 1: Factor Completely
1. 2x2 + 10x – 48
7. 2y5 – 16y4 + 32y3
2. 5y2 – 40y + 60
8. 3x2 - 243
3. 6x3 + 54x2 + 120x
9. 2b3 – 2b2 – 24b
4. 3s3 – 108s
10. 4x3 – 20x2 + 24x
5. 5x4 – 50x3 + 125x2
11. 6w4 – 18w3 – 24w2
6. 2a4 + 12a3 + 18a2
12. 3x2 – 9x - 54
Part 2: Thinking Questions
1. Give at least one example of a trinomial in the form of: x2 +bx +c that can be factored.
2. Give at least one example of a trinomial without a lead coefficient of one that can be factored.
3. Give an example of a prime trinomial.
Factor each trinomial. (3 points each)
1. 3x3 +18x2 – 81x
Copyright © 2009-2010 Karin Hutchinson – Algebra-class.com
2. 2x3 – 8x2 – 8x
Unit 9: Factoring
Lesson 4: Factoring Trinomials (A Special Situation) Answers
Part 1: Factor Completely
1. 2x2 + 10x – 48
Step 1: Since the lead coefficient is greater than
1, see if there is a GCF for all terms.
2 is the GCF for all terms.
2(x2 + 5x – 24)
Step 2: Factor the expression inside of
parenthesis.
We need a positive and negative number that
have a product of -24 and a sum of 5
(8 & 3, 8 must be positive since we need
a +5 for a sum).
Final Answer: 2(x+ 8) (x-3)
7. 2y5 – 16y4 + 32y3
Step 1: Since the lead coefficient is greater than
1, see if there is a GCF for all terms.
3
2y is the GCF for all terms.
2y3(y2 – 8y + 16)
Step 2: Factor the expression inside of
parenthesis.
We need 2 negative numbers that have a product
of 16 and a sum of -8.
(-4 & -4).
2y3(y-4) (y-4)
Final Answer: 2y3(y-4)2
2. 5y2 – 40y + 60
Step 1: Since the lead coefficient is greater than
1, see if there is a GCF for all terms.
5 is the GCF for all terms.
5(y2 - 8x + 12)
Step 2: Factor the expression inside of
parenthesis.
We need 2 negative numbers that have a product
of 12 and a sum of -8
(-6 & -2).
8. 3x2 – 243
Step 1: Since the lead coefficient is greater than
1, see if there is a GCF for all terms.
3 is the GCF for all terms.
3(x2 - 81)
Step 2: Factor the expression inside of
parenthesis.
This is a difference of two squares. So,
what number squared is 81? (9)
Final Answer: 3(x+ 9) (x-9)
Final Answer: 5(y-6) (y-2)
Copyright © 2009-2010 Karin Hutchinson – Algebra-class.com
Unit 9: Factoring
3. 6x3 + 54x2 + 120x
9. 2b3 – 2b2 – 24b
Step 1: Since the lead coefficient is greater than
1, see if there is a GCF for all terms.
6x is the GCF for all terms.
6x(x2 + 9x + 20)
Step 2: Factor the expression inside of
parenthesis.
We need 2 positive numbers that have a product
of 20 and a sum of 9
(4 & 5)
Step 1: Since the lead coefficient is greater than
1, see if there is a GCF for all terms.
2b is the GCF for all terms.
2b(b2 - b – 12)
Step 2: Factor the expression inside of
parenthesis.
We need a positive and negative number that
have a product of -12 and a sum of -1 (4 & 3, 4 must be negative since we need
a -1 for a sum).
Final Answer: 6x(x+ 4) (x+ 5)
Final Answer: 2b(b-4) (b+3)
4. 3s3 – 108s
10. 4x3 – 20x2 + 24x
Step 1: Since the lead coefficient is greater than
1, see if there is a GCF for all terms.
3s is the GCF for all terms.
3s(s2 - 36)
Step 2: Factor the expression inside of
parenthesis.
This is a difference of two squares. What
number when squared equals 36? (6)
Step 1: Since the lead coefficient is greater than
1, see if there is a GCF for all terms.
4x is the GCF for all terms.
4x(x2 – 5x+ 6)
Step 2: Factor the expression inside of
parenthesis.
We need two negative numbers that have
a product of 6 and a sum of -5. (-3 & -2)
Final Answer: 3s(s+ 6) (s-6)
Final Answer: 4x(x-3) (x-2)
5. 5x4 – 50x3 + 125x2
11. 6w4 – 18w3 – 24w2
Step 1: Since the lead coefficient is greater than
1, see if there is a GCF for all terms.
2
5x is the GCF for all terms.
5x2(x2 -10x + 25)
Step 2: Factor the expression inside of
parenthesis.
We need two negative numbers whose
product is 25 and sum is -10 (-5 & -5)
This is a perfect square.
Final Answer: 5x2(x - 5) (x-5) or
Step 1: Since the lead coefficient is greater than
1, see if there is a GCF for all terms.
2
6w is the GCF for all terms.
6w2(w2 -3w -4)
Step 2: Factor the expression inside of
parenthesis.
We need a positive and negative number
whose product is -4 and whose sum is -3.
(-4 & 1) 4 must be negative since the
sum is negative.
5x2(x-5)2
Final Answer: 6w2(w -4) (w +1)
Copyright © 2009-2010 Karin Hutchinson – Algebra-class.com
Unit 9: Factoring
6. 2a4 + 12a3 + 18a2
Step 1: Since the lead coefficient is greater than
1, see if there is a GCF for all terms.
2
2a is the GCF for all terms.
2a2(a2 + 6a + 9)
Step 2: Factor the expression inside of
parenthesis.
This is a perfect square. What number
can you square and get 9, and add to get
6? (3)
12. 3x2 – 9x – 54
Step 1: Since the lead coefficient is greater than
1, see if there is a GCF for all terms.
3 is the GCF for all terms.
3(x2 – 3x - 18)
Step 2: Factor the expression inside of
parenthesis.
What positive and negative numbers have
a product of -18 and a sum of -3? (-6 & 3)
6 must be negative since the sum is
negative.
Final Answer: 2a2(a+3)2
Final Answer: 3(x -6) (x +3)
Part 2: Thinking Questions
1. Give at least one example of a trinomial in the form of: x2 +bx +c that can be factored.
These answers may vary. The easiest way to find a trinomial that can be factored is to work backwards
and start with two binomials. You can choose any binomials.
For example, I’ll choose (x+4) (x+2) now multiply using foil.
(x)(x) +(2)x+(4)x+4(2)
x2 +6x + 8 is a trinomial that can be factored. We know this because we started with the two factors and
multiplied.
2. Give at least one example of a trinomial without a lead coefficient of one that can be factored.
These answers may vary. The easiest way to find a trinomial that can be factored is to work backwards
and start with two binomials. You can choose any binomials. Since you want a lead coefficient that is
greater than 1, make sure that one of your binomials has a coefficient of x that is greater than 1.
For example, I’ll choose (2x+4) (x+2) now multiply using foil.
(2x)(x) +(2x)2+(4)x+4(2)
2x2 +8x + 8 is a trinomial that can be factored. We know this because we started with the two factors and
multiplied.
Copyright © 2009-2010 Karin Hutchinson – Algebra-class.com
Unit 9: Factoring
3. Give an example of a prime trinomial.
A prime trinomial cannot be factored. For example:
x2 + 2x + 2 - We would need two positive numbers whose product is 2 and whose sum is 2. The only way
to make a product of 2 is (2 ●1) Since 2+1 = 3, we know that this cannot be factored.
The best way to make up a prime trinomial is to use a prime number as your constant. Then make sure the
coefficient of x is not the sum of the factors of your prime number.
1. 3x3 +18x2 – 81x
Step 1: Since the lead coefficient is greater than
1, see if there is a GCF for all terms.
3x is the GCF for all terms.
3x(x2 + 6x - 27)
Step 2: Factor the expression inside of
parenthesis.
We need a positive and negative number
whose product is -27 and whose sum is
6. (9 & -3) (9 must be positive since the
sum is positive)
Final Answer: 3x(x+9) (x - 3)
2. 2x3 – 8x2 – 8x
Step 1: Since the lead coefficient is greater than
1, see if there is a GCF for all terms.
2x is the GCF for all terms.
2x(x2- 4x - 4)
Step 2: Factor the expression inside of
parenthesis.
We need a positive and negative number
whose product is -4 and whose sum is -4.
The inside of the parenthesis cannot be
factored any further because there’s no
positive and negative number whose
product is -4 and whose sum is -4.
Final Answer: 2x(x2 – 4x – 4)
Copyright © 2009-2010 Karin Hutchinson – Algebra-class.com