How do factor a trinomial expression with a leading coefficient not equal to 1?

Answer 1

To factor a trinomial in the form ax2 + bx + c, where a does not equal 1, the easiest process is called "factoring by grouping". To factor by grouping, you must change the trinomial into an equivalent tetranomial by rewriting the middle term (bx) as the sum of two terms. There is a specific way to do this, as demonstrated in the example.

Take the quadratic trinomial 5x2 + 11x + 2

1. Find the product of a and c, or 5*2 = 10.

2. Find factors of ac that when added together give you b, in this case 10 and 1.

3. Rewrite the middle term as the sum of the two factors (5x2 + 10x + x + 2).

4. Group terms with common factors and factor these groups.

5x2 + x + 10x + 2

x(5x + 1) + 2(5x + 1)

5. Factor the binomial in the parentheses out of the whole polynomial, leaving you with the product of two binomials. 5x2 + 11x + 2 = (x + 2)(5x + 1)

Notes:

1. The same process is done if there are any minus signs in the trinomial, just be careful when factoring out a negative from a positive or vice versa.

2. If you have a tetranomial on its own, you can skip the rewriting process and just factor the whole polynomial by grouping from the start.

3. As in factoring any polynomial, always factor out the GCF first, then factor the remaining polynomial if necessary.

4. Always look for patterns, like the difference of squares or square of a binomial, while factoring. It will save a lot of time.