Chapter 6 FaCtoring and the distributive ProPerty 143✔SOLUTIONS
- 4x − 10y = 2 · 2x − 2 · 5y = 2(2x − 5y)
- 3x + 6y − 12 = 3 · x + 3 · 2y − 3 · 4 = 3(x + 2y − 4)
- 5x^2 + 15 = 5 · x^2 + 5 · 3 = 5(x^2 + 3)
- 4x^2 + 4x = 4x · x + 4x · 1 = 4x(x + 1)
- 4x^3 − 6x^2 + 12x = 2x · 2x^2 − 2x · 3x + 2x · 6 = 2x(2x^2 − 3x + 6)
- −24xy^2 + 6x^2 + 18x = 6x · (−4y^2 ) + 6x · x + 6x · 3
= 6x(−4y^2 + x + 3) - 30x^4 − 6x^2 = 6x^2 · 5x^2 − 6x^2 · 1 = 6x^2 (5x^2 − 1)
- 15x^3 y^2 z^7 − 30xy^2 z^4 + 6x^4 y^2 z^6 = 3xy^2 z^4 · 5x^2 z^3 − 3xy^2 z^4 · 10 + 3xy^2 z^4 · 2x^3 z^2
= 3xy^2 z^4 (5x^2 z^3 − 10 + 2x^3 z^2 )
Factoring with Negative Numbers
Factoring a negative number from two or more terms has the same effect on
signs within parentheses as distributing a negative number does—every sign
changes. Negative quantities are factored in the next examples and practice
problems.EXAMPLES
Factor a negative quantity.−2 − 3x = −(2 + 3x)x + y
As both terms are positive (that is, have positive 1 as a coefficient), we will
have to use the fact that 1 = (−1)(−1) to write each term with a negative
coefficient. Thus, x + y = (−1)(−1)x + (−1)(−1)y = (−1)[(−1)x + (−1)y] =
(−1)(−x − y) = −(−x – y).
In general, x = −(−x), for any x.
−4 + x = −(4 − x)
2 x^2 + 4x = −2x(−x − 2)
12 xy − 25x = −x(−12y + 25)
x − y − z + 5 = −(−x + y + z − 5)EXAMPLES
Factor a negative quantity.