Lesson 6 · from §53 of the 1910 text
Trinomials $ax^2 + bx + c$
A trinomial ax^2 + bx + c can always be factored by grouping, if you can find two numbers whose product is ac and whose sum is b. If there are none, it's prime.
Example 1. 6x^2 + 17x + 7. Here ac = 42, and of 1 \cdot 42, 2 \cdot 21, 3 \cdot 14, 6 \cdot 7, only 3 and 14 add to 17: 6x^2 + 3x + 14x + 7 = 3x(2x + 1) + 7(2x + 1) = (2x + 1)(3x + 7).
Example 2. 3x^2 + 10x - 8. Here ac = -24, so the numbers have opposite signs: +12 and -2. 3x^2 + 12x - 2x - 8 = 3x(x + 4) - 2(x + 4) = (x + 4)(3x - 2).
Rule. Find two numbers whose product is ac and whose sum is b. Split bx into two terms with those coefficients, then factor by grouping.
Perfect squares and x^2 + bx + c are just special cases of this.
Quick check
2x^2 + 7x + 6
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Factor.
2x^2 + 5x + 3
3x^2 + 5x + 2
2x^2 - 7x + 3
x^2 + 6x + 9
3x^2 + 8x + 4
5x^2 - 8x + 3
9x^2 + 18x + 8
7x^2 + 9ax + 2a^2
4x^2 + 16x + 15
2x^2 - 3x - 2
2x^2 - x - 3
4x^2 - 7x - 15
3x^4 + 7x^2 - 6
3x^2 + 5xy - 2y^2
3x^2 - x - 2
2x^2 - 5x - 3
2x^2 + xz - 3z^2
6x^2 - x - 2
25x^2 - 20x + 3
49x^2 - 21x + 2
36x^2 - 36x + 5
6x^4 + 13x^2 + 6
9x^6 - 6x^3 - 8
5x^2 + 2x - 16
3x^2 - 11x - 20
5x^4 + 18x^2y + 16y^2
6x^2 + 23x - 55
6x^6 + 23x^3 + 21
10x^2 - 7x - 12
4c^2 + 11cg - 3g^2
8x^2 + 26xy + 15y^2
35x^2 + 22x + 3
14x^2 - x - 3
21x^4 - x^2 - 2
22x^2 - 3x - 7
18x^2 + 65x + 7
26x^2 + 9x - 2
14x^2 - 39x + 10
35x^2 - 39x - 36
42x^2 - 9x - 6
Review: factor completely.
4ax^2 + 8ax + 3a
2c^2x^2 + c^2x - 3c^2
48ax^4 - 120ax^2 + 75a
18c^2d - 30cd - 28d
x^4 + x^3 - 110x^2
75x^3 + 60x^2 + 12x
4x^2y^2 - 4xy^2 - 120y^2
70x^3 - 85x^2y - 30xy^2
18x^3 - 39x^2 + 18x
36x^4 - 6x^3 - 12x^2
4x^4 - 20x^3 - 24x^2
5ax^2 + 5ax + 5a