Concept Check An expression is factored when it is written as a product, not a sum. Which
of the following are not factored?
Complete each factoring by writing each polynomial as the product of two factors.
2 k^2 + 15 k+ 12 12 k^2 + 5 k 2 + 1
2 k^215 k 2 2 k^215 k+ 12
366 CHAPTER 6 Factoring and Applications
19. 20. 21.
=- 4 z^512
12 p^5
= 3 m^212
9 m^4
22. 23. 24.
= 9 a^2 b 1 2
27 a^3 b^2
= 3 m^3 n 1 2
6 m^4 n^5
=- 5 k^812
25. 26. 27.
= 10 a 1 2
10 a^2 - 20 a
= 181 2
18 p+ 36
= 121 2
12 y+ 24
28. 29. 30.
= 2 st 1 2
18 s^3 t^2 + 10 st
= 4 x^2 y 1 2
8 x^2 y+ 12 x^3 y^2
= 15 x 1 2
15 x^2 - 30 x
31.How can you check your answer when you factor a polynomial?
32.Concept Check A student factored as WHAT WENT WRONG?
Factor correctly.
Write in factored form by factoring out the greatest common factor. See Examples 3 and 4.
- 12 x^3 + 6 x^2 42. 21 b^3 + 7 b^2 43. 65 y^10 + 35 y^6 44. 100 a^5 + 16 a^3
27 m^3 - 9 m 36 p^3 - 24 p 16 z^4 + 24 z^225 k^4 + 15 k^2
x^2 - 4 x m^2 - 7 m 6 t^2 + 15 t 8 x^2 + 6 x
18 x^3 y^2 + 9 xy 9 xy 12 x^2 y 2.
- 11 w^3 - 100 46. 13 z^5 - 80 47. 8 mn^3 + 24 m^2 n^3 48. 19 p^2 y+ 38 p^2 y^3
Students often have difficulty when factoring by grouping because they are not able to tell
when the polynomial is completely factored. For example,
Not in factored form
is not in factored form, because it is the sumof two terms: and.
However, because is a common factor of these two terms, the expression can now be
factored.
In factored form
The factored form is a productof two factors: and.
Concept Check Determine whether each expression is in factored form or is not in factored
form. If it is not in factored form, factor it if possible.
- 67.Concept Check Why is it not possible to factor the expression in Exercise 65?
18 x^21 y+ 42 + 71 y- 42 12 k^31 s- 32 + 71 s+ 32
18 +x 217 t+ 42 13 r+ 7215 x- 12
817 t+ 42 +x 17 t+ 42 3 r 15 x- 12 + 715 x- 12
2 x- 3 5 y+ 8 t
12 x- 3215 y+ 8 t 2
2 x- 3
5 y 12 x- 32 8 t 12 x- 32
5 y 12 x- 32 + 8 t 12 x- 32
q^21 p- 42 + 11 p- 42 y^2 1 x- 92 + 1 1 x- 92
m 1 m+ 2 n 2 +n 1 m+ 2 n 2 q 1 q+ 4 p 2 +p 1 q+ 4 p 2
c 1 x+ 22 - d 1 x+ 22 r 1 x+ 52 - t 1 x+ 52
a^5 + 2 a^3 b^2 - 3 a^5 b^2 + 4 a^4 b^3 x^6 + 5 x^4 y^3 - 6 xy^4 + 10 xy
36 p^6 q+ 45 p^5 q^4 + 81 p^3 q^2125 a^3 z^5 + 60 a^4 z^4 + 85 a^5 z^2
13 y^8 + 26 y^4 - 39 y^25 x^5 + 25 x^4 - 20 x^3
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