(e)
Sum of cubes= 1 x+ 2 +t 21 x^2 + 4 x+ 4 - xt- 2 t+ t^22 Multiply. NOW TRY
= 31 x+ 22 + t 431 x+ 222 - 1 x+ 22 t+t^24
1 x+ 223 +t^3
SECTION 6.3 Special Factoring 337
NOW TRY
EXERCISE 4
Factor each polynomial.
(a)
(b)
(c) 1 x- 323 +y^3
81 a^6 + 3 b^31000 +z^3NOW TRY ANSWERS
- (a)
(b)(c)3 y+y^221 x^2 - 6 x+ 9 - xy+1 x- 3 +y 2 #
19 a^4 - 3 a^2 b+b^22313 a^2 +b 2 #
1100 - 10 z+z^22110 +z 2 #
CAUTION A common error when factoring or is to think that
the xy-term has a coefficient of 2. Since there is no coefficient of 2, expressions of
the form x^2 +xy+y^2 and x^2 - xy+ y^2 usually cannot be factored further.
x^3 + y^3 x^3 - y^3
The special types of factoring are summarized here. These should be memorized.
Special Types of FactoringDifference of Squares
Perfect Square Trinomial
Difference of Cubes
Sum of Cubes x^3 y^3 1 xy 21 x^2 xyy^22
x^3 y^3 1 xy 21 x^2 xyy^22
x^2 2 xyy^2 1 xy 22
x^2 2 xyy^2 1 xy 22
x^2 y^2 1 xy 21 xy 2
Complete solution available
on the Video Resources on DVD
6.3 EXERCISES
Concept Check Work each problem.
1.Which of the following binomials are differences of squares?
A. B. C. D.
2.Which of the following binomials are sums or differences of cubes?
A. B. C. D.
3.Which of the following trinomials are perfect squares?
A. B.
C. D.
4.Of the 12 polynomials listed in Exercises 1–3,which ones can be factored by the methods
of this section?
5.The binomial is an example of a sum of two squares that can be factored. Under
what conditions can the sum of two squares be factored?
6.Insert the correct signs in the blanks.
(a)
(b) nFactor each polynomial. See Examples 1– 4.
- 64 m^4 - 4 y^4 14. 243 x^4 - 3 t^4 15. 1 y+z 22 - 81
36 m^2 - 25 18 a^2 - 98 b^232 c^2 - 98 d 2p^2 - 16 k^2 - 9 25 x^2 - 4n^3 - 1 = 1 n 121 n 2 128 +m^3 = 12 m 214 2 m m^224 x^2 + 649 z^4 + 30 z^2 + 25 25 p^2 - 45 p+ 81x^2 - 8 x- 16 4 m^2 + 20 m+ 2564 +r^3125 - p^69 x^3 + 125 1 x+y 23 - 164 - k^22 x^2 - 25 k^2 + 9 4 z^4 - 49