Common Factors
Factor completely: 6x² + 9x.
3x(2x + 3).
Factor: x² + 7x + 12.
(x + 3)(x + 4).
For 6x² + 11x + 3, what is the product ac used in the grouping method?
ac = 6 × 3 = 18.
Expand: (x + 5)².
x² + 10x + 25.
True or false? x² + 9 = (x + 3)(x − 3). If false, expand the right-hand side.
False: (x + 3)(x − 3) = x² − 9. The expression x² + 9 is prime over the integers.
Factor completely: 12x³ − 18x².
6x²(2x − 3).
Factor: x² − 9x + 20.
(x − 4)(x − 5).
Which two integers have product −18 and sum −7? These split the middle term of 6x² − 7x − 3.
−9 and 2.
Expand: (2x − 3)².
4x² − 12x + 9.
Correct this expansion and identify the missing term: (x − 3)² = x² + 9.
(x − 3)² = x² − 6x + 9. The missing term is −6x.
Factor completely: −4x² + 12x. Take out a negative common factor.
−4x(x − 3).
Factor: x² + x − 12.
(x + 4)(x − 3).
Complete the grouping step, then factor: 6x² + 9x + 2x + 3 = 3x(2x + 3) + ___.
The blank is 1(2x + 3). Complete factorization: (3x + 1)(2x + 3).
Factor: x² − 49.
(x − 7)(x + 7).
Solve by factoring: x² − 5x + 6 = 0.
(x − 2)(x − 3) = 0, so x = 2 or x = 3.
Factor completely over the integers: 3x² + 21x + 30.
3(x + 5)(x + 2).
Factor: 2x² + 7x + 3.
(2x + 1)(x + 3).
Factor 8x² + 2x − 3 by grouping. State the two numbers used to split the middle term.
Split 2x into 6x − 4x. Factorization: (4x + 3)(2x − 1).
Factor: 9x² − 24x + 16.
(3x − 4)².
Factor completely: (x + 1)² + 5(x + 1) + 6.
Let u = x + 1. Then (u + 2)(u + 3) = (x + 3)(x + 4).
Factor completely: −6x² + 9x + 6.
−3(2x + 1)(x − 2).
Factor: 6x² − 13x + 6.
(3x − 2)(2x − 3).
Factor 10x² − 9x + 2 by grouping. Show your groups.
10x² − 5x − 4x + 2 = 5x(2x − 1) − 2(2x − 1) = (5x − 2)(2x − 1).
Factor completely: 8x³ − 27.
(2x − 3)(4x² + 6x + 9), using the difference of cubes.
Factor completely over the integers: x⁴ − 5x² + 4.
Let u = x². Then (u − 1)(u − 4) = (x² − 1)(x² − 4) = (x − 1)(x + 1)(x − 2)(x + 2).