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FACTORING A TRINOMIAL
To factor means to rewrite a single expression as a multiplication problem.
For example, when we're asked to factor the number 12, people respond with answers such as (2) times (6), (3) times (4), or (1) times (12).
"12" is a single expression; (2) times (6) is a multiplication problem written as (2)(6). So one way to "factor" 12 is to write it as (2) times (6). This is factoring.
When we "factor" x^2 + 5x + 6 as ( x + 2 ) times ( x + 3 ), we are rewriting the single expression (the polynomial: x^2 + 5x + 6 ) as a multiplication problem, ( x + 2 )( x + 3 ).
Methods of Factoring Polynomials: Trinomials.
Factor: x^2 - 7x + 12
Goals: We must answer 3 questions to factor any trinomial (a polynomial with 3 terms):
- "What times what" will create the first term of the polynomial? These become the candidates for the first terms of each factor: ( First 1 .........)( First 2 .........)
- "What times what" will create the last term? These become the last term candidates: (......... Last 1 )(......... Last 2 )
- Next multiply the two nearest terms (the inner terms). Then multiply the two most distant terms (the outer terms). Add the two answers. If the factors are correct, the result should be the second term in the original polynomial.
In our example, x^2 - 7x + 12
- "What times what" will create x^2?
Answer: (x) times (x).
So we write ( x .......) ( x ......)
- "What times what" will yield 12?
Answer: ( 1)(12), (2)(6), or (3)(4).
Looking closely at the choices, which pair is most likely to produce the middle term of 7x in the original polynomial? Probably (3)(4), so we'd have: ( x ... 3 )( x ... 4 ). We'll check it in the next step.
- Add the inner and outer products. The 3 in the first parentheses and the x in the second are the two nearest terms. When multiplied, they produce 3x called the " inner product ".
The x in the first parentheses and the 4 in the second parentheses are the farthest terms apart. Their product is 4x called the " outer product ".
Adding 3x and 4x produces 7x , but we want the result to be negative. This will occur when both the 3 and the 4 are negative. So we have: x^2 - 7x + 12 = (x - 3)(x - 4)
We have factored the polynomial since it is now written as a multiplication problem.
ANOTHER EXAMPLE:
Factor: 24x^2 - 14xy - 3y^2
1." What times what" creates 24x^2? Answers: (1x)(24x), (2x)(12x), (3x)(8x), (4x)(6x).
So when we write the first terms in the factors they might look like: (1x......)(24x.....) or (2x.....)(12x.....) or .....
- "What times what" creates 3y^2?
Answers: (1y)(3y)
The last terms in the factors will be: (......1y)(......3y)
- What is the sum of the inner and outer products?
Here we have to be creative, use trial and error, and some arithmetic. If we pick the factors: (1x 1y)(24x 3y), the inner product is (1y)(24x) = 24xy and the outer product is (1x)(3y) = 3xy. The sum of 24xy + 3xy is 27xy. We want -14xy (see the second term of the original polynomial). If one term was negative and one was positive, we'd still get -24xy + 3xy = -21xy (this is the trial and error part).
