By comparing the coefficients of the terms in the two trinomials, we see that \(\alert{a + b = 10}\) and \(\blert{ab = 16}\text{.}\) That is, the sum of the two numbers is the coefficient of the linear term, \(10\text{,}\) and their product is the constant term, \(16\text{.}\)
To find the numbers, we list all the possible integer factorizations of \(16\text{:}\)
In Example A.57 we factor quadratic trinomials in which one or more of the coefficients is negative.
ExampleA.57.
Factor.
\(\displaystyle x^2-7x+12\)
\(\displaystyle x^2-x-12\)
Solution.
Find two numbers whose product is \(12\) and whose sum is \(-7\text{.}\) Because the product is positive and the sum is negative, the two numbers must both be negative. The possible factors of \(12\) are \(-1\) and \(-12\text{,}\)\(-2\) and \(-6\text{,}\) or \(-3\) and \(-4\text{.}\) Only \(-4\) and \(-3\) have the correct sum, \(-7\text{.}\) Hence,
Find two numbers whose product is \(-12\) and whose sum is \(-1\text{.}\) Because the product is negative, the two numbers must be of opposite sign and their sum must be \(-1\text{.}\) By listing the possible factors of \(-12\text{,}\) we find that the two numbers are \(-4\) and \(3\text{.}\) Hence,
Consider possible factors for the constant term: \(9\) may be factored as \(9\cdot 1\) or as \(3\cdot 3\text{.}\) Form all possible pairs of binomial factor using these factorizations.
Select the combinations of the products \(\Large{①}\) and \(\Large{②}\) whose sum or difference could be the linear term, \(-21x\text{.}\)
With practice, you can usually factor trinomials of the form \(Ax^2 + Bx + C\) mentally. The following observations may help.
If \(A\text{,}\)\(B\) and \(C\) are all positive, both signs in the factored form are positive. For example, as a first step in factoring \(6x^2 + 11x + 4\text{,}\) we could write
If \(A\) and \(C\) are positive and \(B\) is negative, both signs in the factored form are negative. Thus as a first step in factoring \(6x^2 - 11x + 4\text{,}\) we could write
Each of the formulas for special products, when viewed from right to left, also represents a special case of factoring quadratic polynomials.
Special Factorizations.
\(\displaystyle a^2 + 2ab + b^2=(a + b)^2\)
\(\displaystyle a^2 - 2ab + b^2=(a - b)^2 \)
\(\displaystyle a^2 - b^2=(a + b)(a-b)\)
\(\displaystyle a^2+b^2 ~~\text{ cannot be factored}\)
The trinomials in (I) and (II) are sometimes called perfect-square trinomials because they are squares of binomials. Note that the sum of two squares, \(a^2 + b^2\text{,}\) cannot be factored.
ExampleA.62.
Factor.
\(\displaystyle x^2 + 8x + 16\)
\(\displaystyle y^2 -10y + 25\)
\(\displaystyle 4a^2 - 12ab + 9b^2\)
\(\displaystyle 25m^2n^2 + 20mn + 4\)
Solution.
Because \(16\) is equal to \(4^2\) and \(8\) is equal to \(2\cdot 4\text{,}\)
Binomials of the form \(a^2 - b^2\) are often called the difference of two squares.
ExampleA.63.
Factor if possible.
\(\displaystyle x^2 - 81\)
\(\displaystyle 4x^2 - 9y^2\)
\(\displaystyle x^2 + 81\)
Solution.
The expression \(x^2 - 81\) is the difference of two squares, \(x^2 - 9^2\text{,}\) and thus can be factored according to Special Factorization (III) above.
The expression \(x^2 + 81\text{,}\) or \(x^2 + 0x + 81\text{,}\) is not factorable, because no two real numbers have a product of \(81\) and a sum of \(0\text{.}\)
CautionA.64.
\(x^2 + 81\ne (x + 9) (x + 9)\text{,}\) which you can verify by multiplying
The factors \(x + 9\) and \(x - 9\) in Example A.63a are called conjugates of each other. In general, any binomials of the form \(a+b\) and \(a-b\) are called a conjugate pair.
SubsectionSection Summary
SubsubsectionVocabulary
Look up the definitions of new terms in the Glossary.
Perfect-square trinomial
Difference of squares
Conjugate
SubsubsectionSKILLS
Practice each skill in the exercises listed.
Factor quadratic trinomials: #1–36
Expand special products: #37–48
Factor special quadratic expressions: #49–68
ExercisesExercises A.8
Exercise Group.
For Problems 1-36, factor completely.
1.
\(x^2+5x+6 \)
2.
\(x^2+5x+4\)
3.
\(y^2-7y+12\)
4.
\(y^2-7y+10\)
5.
\(x^2-6-x\)
6.
\(x^2-15-2x\)
7.
\(2x^2+3x-2\)
8.
\(3x^2-7x+2\)
9.
\(7x+4x^2-2\)
10.
\(1-5x+6x^2\)
11.
\(9y^2-21y-8\)
12.
\(10y^2-3y-18\)
13.
\(10u^2-3-u\)
14.
\(8u^2-3+5u\)
15.
\(21x^2-43x-14\)
16.
\(24x^2-29x+5\)
17.
\(5a+72a^2-25\)
18.
\(-30a+72a^2-25\)
19.
\(12-53x+30x^2\)
20.
\(39x+80x^2-20\)
21.
\(-30t-44+54t^2\)
22.
\(48t^2-122t+39\)
23.
\(3x^2-7ax+2a^2\)
24.
\(9x^2+9ax-10a^2\)
25.
\(15x^2-4xy-4y^2\)
26.
\(12x^2+7xy-12y^2\)
27.
\(18u^2+20v^2-39uv\)
28.
\(24u^2-20v^2+17uv\)
29.
\(12a^2-14b^2-13ab\)
30.
\(24a^2-15b^2-2ab\)
31.
\(10a^2b^2-19ab+6\)
32.
\(12a^2b^2-ab-20\)
33.
\(56x^2y^2-2xy-4\)
34.
\(54x^2y^2+3xy-2\)
35.
\(22a^2z^2-21-19az\)
36.
\(26a^2z^2-24+23az\)
Exercise Group.
For Problems 37-48, write the expression as a polynomial and simplify.
