Learning Objectives
By the end of this section, you will be able to:
Recognize and use the appropriate method to factor a polynomial completely
Recognize and Use the Appropriate Method to Factor a Polynomial Completely
You have now become acquainted with all the methods of factoring that you will need in this course. The following chart summarizes all the factoring methods we have covered, and outlines a strategy you should use when factoring polynomials.
General Strategy for Factoring Polynomials
How To
Use a general strategy for factoring polynomials.
Step 1. Is there a greatest common factor?Factor it out.
Step 2. Is the polynomial a binomial, trinomial, or are there more than three terms?If it is a binomial: Is it a sum?Of squares? Sums of squares do not factor.Of cubes? Use the sum of cubes pattern. Is it a difference?Of squares? Factor as the product of conjugates.Of cubes? Use the difference of cubes pattern. If it is a trinomial: Is it of the form x2+bx+c?x2+bx+c? Undo FOIL. Is it of the form ax2+bx+c?ax2+bx+c?If a and c are squares, check if it fits the trinomial square pattern.Use the trial and error or “ac” method. If it has more than three terms: Use the grouping method.
Step 3. Check.Is it factored completely?Do the factors multiply back to the original polynomial?
Remember, a polynomial is completely factored if, other than monomials, its factors are prime!
Example 6.35
Factor completely: 7x3−21x2−70x.7x3−21x2−70x.
Solution
row: 7x3−21x2−70x7x3−21x2−70x
row: Is there a GCF? Yes, 7x7x.
row: Factor out the GCF. | 7x(x2−3x−10)7x(x2−3x−10)
row: In the parentheses, is it a binomial, trinomial, or are there more terms?
row: Trinomial with leading coefficient 1.
row: “Undo” FOIL. | 7x(x)(x)7x(x)(x)
row: 7x(x+2)(x−5)7x(x+2)(x−5)
row: Is the expression factored completely? Yes.
row: Neither binomial can be factored.
row: Check your answer.
row: Multiply.
row: 7x(x+2)(x−5)7x(x+2)(x−5)
row: 7x(x2−5x+2x−10)7x(x2−5x+2x−10)
row: 7x(x2−3x−10)7x(x2−3x−10)
row: 7x3−21x2−70x✓7x3−21x2−70x✓
Try It 6.69
Factor completely: 8y3+16y2−24y.8y3+16y2−24y.
Try It 6.70
Factor completely: 5y3−15y2−270y.5y3−15y2−270y.
Be careful when you are asked to factor a binomial as there are several options!
Example 6.36
Factor completely: 24y2−150.24y2−150.
Solution
row: 24y2−15024y2−150
row: Is there a GCF? Yes, 6.
row: Factor out the GCF. | 6(4y2−25)6(4y2−25)
row: In the parentheses, is it a binomial, trinomial or are there more than three terms? Binomial.
row: Is it a sum? No.
row: Is it a difference? Of squares or cubes? Yes, squares. | 6((2y)2−(5)2)6((2y)2−(5)2)
row: Write as a product of conjugates. | 6(2y−5)(2y+5)6(2y−5)(2y+5)
row: Is the expression factored completely?Is the expression factored completely?
row: Neither binomial can be factored.Neither binomial can be factored.
row: Check:
row: Multiply.Multiply.
row: 6(2y−5)(2y+5)6(2y−5)(2y+5)
row: 6(4y2−25)6(4y2−25)
row: 24y2−150✓24y2−150✓
Try It 6.71
Factor completely: 16x3−36x.16x3−36x.
Try It 6.72
Factor completely: 27y2−48.27y2−48.
The next example can be factored using several methods. Recognizing the trinomial squares pattern will make your work easier.
Example 6.37
Factor completely: 4a2−12ab+9b2.4a2−12ab+9b2.
Solution
row: 4a2−12ab+9b24a2−12ab+9b2
row: Is there a GCF? No.
row: Is it a binomial, trinomial, or are there more terms?
row: Trinomial with a≠1a≠1. But the first term is a perfect square.
row: Is the last term a perfect square? Yes. | (2a)2−12ab+(3b)2(2a)2−12ab+(3b)2
row: Does it fit the pattern, a2−2ab+b2a2−2ab+b2? Yes. | (2a)2↘−12ab+−2(2a)(3b)↙(3b)2(2a)2↘−12ab+−2(2a)(3b)↙(3b)2
row: Write it as a square. | (2a−3b)2(2a−3b)2
row: Is the expression factored completely? Yes.Is the expression factored completely? Yes.
row: The binomial cannot be factored.The binomial cannot be factored.
row: Check your answer.
row: Multiply.Multiply.
row: (2a−3b)2(2a−3b)2
row: (2a)2−2·2a·3b+(3b)2(2a)2−2·2a·3b+(3b)2
row: 4a2−12ab+9b2✓4a2−12ab+9b2✓
Try It 6.73
Factor completely: 4x2+20xy+25y2.4x2+20xy+25y2.
Try It 6.74
Factor completely: 9x2−24xy+16y2.9x2−24xy+16y2.
Remember, sums of squares do not factor, but sums of cubes do!
Example 6.38
Factor completely 12x3y2+75xy2.12x3y2+75xy2.
Solution
row: 12x3y2+75xy212x3y2+75xy2
row: Is there a GCF? Yes, 3xy23xy2.
row: Factor out the GCF. | 3xy2(4x2+25)3xy2(4x2+25)
row: In the parentheses, is it a binomial, trinomial, or are there more than three terms? Binomial.
row: Is it a sum? Of squares? Yes. | Sums of squares are prime.
row: Is the expression factored completely? Yes.Is the expression factored completely? Yes.
row: Check:
row: Multiply.Multiply.
row: 3xy2(4x2+25)3xy2(4x2+25)
row: 12x3y2+75xy2✓12x3y2+75xy2✓
Try It 6.75
Factor completely: 50x3y+72xy.50x3y+72xy.
Try It 6.76
Factor completely: 27xy3+48xy.27xy3+48xy.
ਜਦੋਂ ਘਣਾਂ ਦੇ ਜੋੜ ਜਾਂ ਘਟਾਓ ਦੇ ਨਮੂਨੇ ਦੀ ਵਰਤੋਂ ਕਰਦੇ ਹੋ, ਤਾਂ ਨਿਸ਼ਾਨਾਂ ਦਾ ਧਿਆਨ ਰੱਖਣਾ।
ਉਦਾਹਰਨ 6.39
ਪੂਰੀ ਤਰ੍ਹਾਂ ਅਭਾਜ ਗੁਣਨਖੰਡ ਬਣਾਓ: 24x³+81y³.
ਹੱਲ
ਸਤਰ: ਕੀ ਕੋਈ ਸਾਂਝਾ ਗੁਣਨਖੰਡ ਹੈ? ਹਾਂ, 3।
ਸਤਰ: ਇਸਨੂੰ ਬਾਹਰ ਕੱਢੋ।
ਸਤਰ: ਬਰੈਕਟਾਂ ਵਿੱਚ, ਕੀ ਇਹ ਦੋ-ਪਦੀ, ਤਿੰਨ-ਪਦੀ, ਜਾਂ ਤਿੰਨ ਤੋਂ ਵੱਧ ਪਦ ਹਨ? ਦੋ-ਪਦੀ।
ਸਤਰ: ਕੀ ਇਹ ਜੋੜ ਹੈ ਜਾਂ ਘਟਾਓ? ਜੋੜ।
ਸਤਰ: ਵਰਗਾਂ ਦਾ ਜਾਂ ਘਣਾਂ ਦਾ? ਘਣਾਂ ਦਾ ਜੋੜ।
ਸਤਰ: ਇਸਨੂੰ ਘਣਾਂ ਦੇ ਜੋੜ ਦੇ ਨਮੂਨੇ ਦੀ ਵਰਤੋਂ ਕਰਕੇ ਲਿਖੋ।
ਸਤਰ: ਕੀ ਇਹ ਅਭਾਜ ਗੁਣਨਖੰਡ ਬਣ ਗਿਆ ਹੈ? ਹਾਂ।
ਸਤਰ: ਗੁਣਾ ਕਰਕੇ ਜਾਂਚ ਕਰੋ।
ਪ੍ਰਯਾਸ ਕਰੋ 6.77
ਪੂਰੀ ਤਰ੍ਹਾਂ ਅਭਾਜ ਗੁਣਨਖੰਡ ਬਣਾਓ: 250m³+432n³.
ਪ੍ਰਯਾਸ ਕਰੋ 6.78
ਪੂਰੀ ਤਰ੍ਹਾਂ ਅਭਾਜ ਗੁਣਨਖੰਡ ਬਣਾਓ: 2p³+54q³.
ਉਦਾਹਰਨ 6.40
ਪੂਰੀ ਤਰ੍ਹਾਂ ਅਭਾਜ ਗੁਣਨਖੰਡ ਬਣਾਓ: 3x⁵y−48xy.
ਹੱਲ
ਸਤਰ: 3x⁵y−48xy
ਸਤਰ: ਕੀ ਕੋਈ ਸਾਂਝਾ ਗੁਣਨਖੰਡ ਹੈ? 3xy ਨੂੰ ਬਾਹਰ ਕੱਢੋ। | 3xy(x⁴−16)
ਸਤਰ: ਕੀ ਦੋ-ਪਦੀ ਜੋੜ ਜਾਂ ਘਟਾਓ ਹੈ? ਵਰਗਾਂ ਜਾਂ ਘਣਾਂ ਦਾ? ਇਸਨੂੰ ਵਰਗਾਂ ਦੇ ਘਟਾਓ ਵਜੋਂ ਲਿਖੋ। | 3xy((x²)²−(4)²)
ਸਤਰ: ਇਸਨੂੰ ਸੰਯੁਗਮਾਂ ਦੇ ਗੁਣਨਫਲ ਵਜੋਂ ਅਭਾਜ ਗੁਣਨਖੰਡ ਬਣਾਓ। | 3xy(x²−4)(x²+4)
ਸਤਰ: ਪਹਿਲੀ ਦੋ-ਪਦੀ ਫਿਰ ਤੋਂ ਵਰਗਾਂ ਦਾ ਘਟਾਓ ਹੈ। | 3xy((x)²−(2)²)(x²+4)
ਸੰਯੁਗਮਾਂ ਦੇ ਗੁਣਨਫਲ ਵਜੋਂ ਇਸਨੂੰ ਕਾਰਕੀਬੱਧ ਕਰੋ।
ਕੀ ਇਹ ਵਿਅੰਜਨ ਪੂਰੀ ਤਰ੍ਹਾਂ ਕਾਰਕੀਬੱਧ ਹੈ? ਹਾਂ।
ਆਪਣੇ ਜਵਾਬ ਦੀ ਜਾਂਚ ਕਰੋ।
ਗੁਣਾ ਕਰੋ।
3xy(x−2)(x+2)(x2+4)
3xy(x2−4)(x2+4)
3xy(x4−16)
3x5y−48xy✓
ਇਸਨੂੰ ਅਜ਼ਮਾਓ 6.79
ਪੂਰੀ ਤਰ੍ਹਾਂ ਕਾਰਕੀਬੱਧ ਕਰੋ: 4a5b−64ab.
ਇਸਨੂੰ ਅਜ਼ਮਾਓ 6.80
ਪੂਰੀ ਤਰ੍ਹਾਂ ਕਾਰਕੀਬੱਧ ਕਰੋ: 7xy5−7xy.
ਉਦਾਹਰਨ 6.41
ਪੂਰੀ ਤਰ੍ਹਾਂ ਕਾਰਕੀਬੱਧ ਕਰੋ: 4x2+8bx−4ax−8ab.
ਹੱਲ
4x2+8bx−4ax−8ab
ਕੀ ਕੋਈ ਸਾਂਝਾ ਗੁਣਨਖੰਡ ਹੈ? ਸਾਂਝਾ ਗੁਣਨਖੰਡ, 4, ਬਾਹਰ ਕੱਢੋ।
ਚਾਰ ਪਦ ਹਨ। ਸਮੂਹੀਕਰਨ ਦੀ ਵਰਤੋਂ ਕਰੋ।
ਕੀ ਇਹ ਵਿਅੰਜਨ ਪੂਰੀ ਤਰ੍ਹਾਂ ਕਾਰਕੀਬੱਧ ਹੈ? ਹਾਂ।
ਆਪਣੇ ਜਵਾਬ ਦੀ ਜਾਂਚ ਕਰੋ। ਗੁਣਾ ਕਰੋ।
ਇਸਨੂੰ ਅਜ਼ਮਾਓ 6.81
ਪੂਰੀ ਤਰ੍ਹਾਂ ਕਾਰਕੀਬੱਧ ਕਰੋ: 6x2−12xc+6bx−12bc.
ਇਸਨੂੰ ਅਜ਼ਮਾਓ 6.82
ਪੂਰੀ ਤਰ੍ਹਾਂ ਕਾਰਕੀਬੱਧ ਕਰੋ: 16x2+24xy−4x−6y.
Taking out the complete GCF in the first step will always make your work easier.
Example 6.42
Factor completely: 40x2y+44xy−24y.40x2y+44xy−24y.
Solution
row: 40x2y+44xy−24y40x2y+44xy−24y
row: Is there a GCF? Factor out the GCF, 4y4y. | 4y(10x2+11x−6)4y(10x2+11x−6)
row: Factor the trinomial with a≠1a≠1. | 4y(10x2+11x−6)4y(10x2+11x−6)
row: 4y(5x−2)(2x+3)4y(5x−2)(2x+3)
row: Is the expression factored completely? Yes.
row: Check your answer.
row: Multiply.
row: 4y(5x−2)(2x+3)4y(5x−2)(2x+3)
row: 4y(10x2+11x−6)4y(10x2+11x−6)
row: 40x2y+44xy−24y✓40x2y+44xy−24y✓
Try It 6.83
Factor completely: 4p2q−16pq+12q.4p2q−16pq+12q.
Try It 6.84
Factor completely: 6pq2−9pq−6p.6pq2−9pq−6p.
When we have factored a polynomial with four terms, most often we separated it into two groups of two terms. Remember that we can also separate it into a trinomial and then one term.
Example 6.43
Factor completely: 9x2−12xy+4y2−49.9x2−12xy+4y2−49.
Solution
row: 9x2−12xy+4y2−499x2−12xy+4y2−49
row: Is there a GCF? No.
row: With more than 3 terms, use grouping. Last 2 terms have no GCF. Try grouping first 3 terms. | 9x2−12xy+4y2−499x2−12xy+4y2−49
row: Factor the trinomial with a≠1a≠1. But the first term is a perfect square.
row: Is the last term of the trinomial a perfect square? Yes. | (3x)2−12xy+(2y)2−49(3x)2−12xy+(2y)2−49
row: Does the trinomial fit the pattern, a2−2ab+b2a2−2ab+b2? Yes. | (3x)2↘−12xy+−2(3x)(2y)↙(2y)2−49(3x)2↘−12xy+−2(3x)(2y)↙(2y)2−49
row: Write the trinomial as a square. | (3x−2y)2−49(3x−2y)2−49
row: Is this binomial a sum or difference? Of squares or cubes? Write it as a difference of squares. | (3x−2y)2−72(3x−2y)2−72
row: Write it as a product of conjugates. | ((3x−2y)−7)((3x−2y)+7)((3x−2y)−7)((3x−2y)+7)
row: (3x−2y−7)(3x−2y+7)(3x−2y−7)(3x−2y+7)
row: Is the expression factored completely? Yes.
row: Check your answer.
row: Multiply.
row: (3x−2y−7)(3x−2y+7)(3x−2y−7)(3x−2y+7)
row: 9x2−6xy−21x−6xy+4y2+14y+21x−14y−499x2−6xy−21x−6xy+4y2+14y+21x−14y−49
row: 9x2−12xy+4y2−49✓9x2−12xy+4y2−49✓
Try It 6.85
Factor completely: 4x2−12xy+9y2−25.4x2−12xy+9y2−25.
Try It 6.86
Factor completely: 16x2−24xy+9y2−64.16x2−24xy+9y2−64.
Practice Makes Perfect
Recognize and Use the Appropriate Method to Factor a Polynomial Completely
In the following exercises, factor completely.
2 n 2 + 13 n − 7 2 n 2 + 13 n − 7
8 x 2 − 9 x − 3 8 x 2 − 9 x − 3
a 5 + 9 a 3 a 5 + 9 a 3
75 m 3 + 12 m 75 m 3 + 12 m
121 r 2 − s 2 121 r 2 − s 2
49 b 2 − 36 a 2 49 b 2 − 36 a 2
8 m 2 − 32 8 m 2 − 32
36 q 2 − 100 36 q 2 − 100
25 w 2 − 60 w + 36 25 w 2 − 60 w + 36
49 b 2 − 112 b + 64 49 b 2 − 112 b + 64
m 2 + 14 m n + 49 n 2 m 2 + 14 m n + 49 n 2
64 x 2 + 16 x y + y 2 64 x 2 + 16 x y + y 2
7 b 2 + 7 b − 42 7 b 2 + 7 b − 42
30 n 2 + 30 n + 72 30 n 2 + 30 n + 72
3 x 4 y − 81 x y 3 x 4 y − 81 x y
4 x 5 y − 32 x 2 y 4 x 5 y − 32 x 2 y
k 4 − 16 k 4 − 16
m 4 − 81 m 4 − 81
5 x 5 y 2 − 80 x y 2 5 x 5 y 2 − 80 x y 2
48 x 5 y 2 − 243 x y 2 48 x 5 y 2 − 243 x y 2
15 p q − 15 p + 12 q − 12 15 p q − 15 p + 12 q − 12
12 a b − 6 a + 10 b − 5 12 a b − 6 a + 10 b − 5
4 x 2 + 40 x + 84 4 x 2 + 40 x + 84
5 q 2 − 15 q − 90 5 q 2 − 15 q − 90
4 u 5 + 4 u 2 v 3 4 u 5 + 4 u 2 v 3
5 m 4 n + 320 m n 4 5 m 4 n + 320 m n 4
4 c 2 + 20 c d + 81 d 2 4 c 2 + 20 c d + 81 d 2
25 x 2 + 35 x y + 49 y 2 25 x 2 + 35 x y + 49 y 2
10 m 4 − 6250 10 m 4 − 6250
3 v 4 − 768 3 v 4 − 768
36 x 2 y + 15 x y − 6 y 36 x 2 y + 15 x y − 6 y
60 x 2 y − 75 x y + 30 y 60 x 2 y − 75 x y + 30 y
8 x 3 − 27 y 3 8 x 3 − 27 y 3
64 x 3 + 125 y 3 64 x 3 + 125 y 3
y 6 − 1 y 6 − 1
y 6 + 1 y 6 + 1
9 x 2 − 6 x y + y 2 − 49 9 x 2 − 6 x y + y 2 − 49
16 x 2 − 24 x y + 9 y 2 − 64 16 x 2 − 24 x y + 9 y 2 − 64
( 3 x + 1 ) 2 − 6 ( 3 x + 1 ) + 9 ( 3 x + 1 ) 2 − 6 ( 3 x + 1 ) + 9
( 4 x − 5 ) 2 − 7 ( 4 x − 5 ) + 12 ( 4 x − 5 ) 2 − 7 ( 4 x − 5 ) + 12
Writing Exercises
Explain what it mean to factor a polynomial completely.
The difference of squares y4−625y4−625 can be factored as (y2−25)(y2+25).(y2−25)(y2+25). But it is not completely factored. What more must be done to completely factor.
Of all the factoring methods covered in this chapter (GCF, grouping, undo FOIL, ‘ac’ method, special products) which is the easiest for you? Which is the hardest? Explain your answers.
Create three factoring problems that would be good test questions to measure your knowledge of factoring. Show the solutions.
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?