LLEVADA’S ALGEBRA 1
Section 7.1
Common Factors in Polynomials
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The FACTORS of numbers are always smaller than the numbers.
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In mathematics, factors are the parts that make a whole, and factoring involves division because they are the
numbers you get when you divide a number or expression (a given quantity) without a remainder.
--- = 3 , the factors of 6 are 2 and 3.
Example: If we divide 6
2
To find ALL the factors of a number (called prime factorization), divide the number by each prime number
starting with the number 2, until you can’t divide again without getting a remainder (decimal.)
Example:
Find all the factors of 54
54
------ = 27
------ = 9
2
--- = 3
3
3
3
Example:
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Answer: The factors of 54 are (2)(3)(3)(3) =  2   3 
Factor 18 + 36 + 54 completely
The prime factorization of 18 is (2)(3)(3)
The prime factorization of 36 is (2)(2)(3)(3)
The prime factorization of 54 is (2)(3)(3)(3)
2 is common to all three numbers once: 2
3 is common to all three numbers twice: (3)(3) = 9
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(9)(2) = 18
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Because the highest common factor for all three numbers is 18, the expression 18 + 36 + 54 can be factored
to:
18(1 + 2 + 3)
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18 36
54
By dividing ------ , ------ , and ------ , 18 has been factored to show that 18 + 36 + 54 can also be written as the prod18 18
18
uct of
18(1 + 2 + 3)
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In algebra, factoring works the same way, except this time with numbers AND symbols.
Example:
Factor 18a3 + 36a2 + 54a completely
The factors of of 18a3 are (2)(3)(3)(a)(a)(a)
The factors of of 36a2 are (2)(2)(3)(3)(a)(a)
The factors of of 54a2 are (2)(3)(3)(3)(a)
Common in all three terms we find (2)(3)(3)(a) = 18a
Because the highest common factor for all three numbers is 18a, the expression 18a3 + 36a2 + 54a can be
factored:
18a (a2 + 2a + 3)
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Chapter 7: Factoring
LLEVADA’S ALGEBRA 1
3
2
18a
36a
54a
By dividing ----------- = a2, ----------- = 2a, and --------- = 3, 18a has been factored to show that:
18a
18a
18a
18a3 + 36a2 + 54a
Factor completely
6
4
3
4x + 6x – 14x + 2x
18a (a2 + 2a + 3)
2
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Example:
can also be written as the product of
The common factor is then 2x2.
By dividing each term of the expression by 2x2
2x2 (2x4 + 3x2 – 7x + 1)
Practice:
Factor completely.
1. 3x3 + 6x2 + 12x
4x6 becomes 2x4
6x4 becomes 3x2
14x3 becomes 7x
2x2 becomes 1
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Answer:
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The common coefficient found in 4, 6, 14, and 2 is 2.
The common base found x.
The common exponent found is 2.
20. 5j3k3 + 10j2k8 – 5j6k7 + 17j7k5
2. 5x4 – 15x2 + 25x
21. 6x7y7z7 + 9x8y7z6 – 9x5y4z11 + 18
3. 12x3y3 – 8x2y2 + 4xy
4. 3x2y2 – 6x3y3 – 9x2y2
5. 2x6 + 4x5 – 6x4 + 8x3
22. 20m2n4 + 16m3n5 – 12m7n8 + 4m7n5
23. 40x2y3z4 – 30x5y5z5 + 20x5y7z10 – 10y7z8
24. 21p6q2 + 24p2q4 – 14p5q9 + 7p3q8
25. 24x3y3z3 + 12x9y8z7 + 18x2y3z4 – 6
7. z3 + z4 + z5 + z7
26. 60s6t6 – 15s4t4 – 45s5t5 + 15s3t7
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6. 5y5 – 10y4 – 15y3 + 20y2
27. x4y5z6 + x7y7z7 – x7y8z9 + x6y7z8
9. 3a4b4 + 6a3b3 – 9a2b2 – 12ab
28. 14u8v12 + 10u9v8 + 6u7v6 + 2u5v3
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8. 4a4 – 8a3 – 12a2 + 16a
29. 4x12y12z12 – 8x8y6z2 – 4x8y6z2 – 4x8y6z2
11. 6x4y4z4 – 12x6y4z3 – 18x6y4z3 + 24xyz
30. 18a14b18 – 12a10b3 – 6a3b9 + 6a7b2
12. 2a2b2 + 6a4b4 + 8a5b5 – 14a3b5
31. 3x4y5z6 + 4x5y14z8 + 6x7y15z8 + 8
13. 3x6y3z6 + 5x7y3z5 – 4x3y3z3 + 8xyz
32. 2c3d3 – 5c3d3 – 6c2d2 – 8cd
14. 3c5d5 – 6c7d7 + 9c6d6 + 18c8d8
33. 6x4y8z3 + 9x7y5z3 + 12x6y7z8 + 15x3y4z5
15. 5x8y7z + 5x5y4z3 – 15x4y3z2 – 35
34. 17e2f + 34e3f3 – 17e2f4 + 51e2f
16. 20e5f + 18e6f8 – 10e7f4 + 36
35. 5x3y4z9 + 5x3y8z7 + 5xy3z4 – 5
17. 5x3y3z3 – 15x5y4z3 + 10x2y9z8 + 5x2y5z7
36. 7g2h – 2g9h7 – 7g3h9 + 14g7h4
18. 21g8h5 + 7g5h6 – 14g7h8 + 28g8h9
37. 8xy5z + 16x2yz3 – 14x4y5z4 + 28x2y5z8
19. 21x6y6z7 – 9x3y2z5 + 6x7y7z7 – 3x2y3z
38. 9x3y5z – 36xy3z6 + 27x2y7z6 – 18
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10. 10a5b5c5 + 15a3b3c3 – 20a2b2c2 + 25abc
7.1 Common Factors in Polynomials
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