How to Factor Trinomials

How to Factor Trinomials

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The Way to Factor a Trinomial in Three Simple Steps

Being able to factor a trinomial is an exceedingly vital and worthwhile skill if you are doing algebra, however, this process may additionally be quite troublesome.

Definition of Trinomials

Trinomials are polynomials with \(3\) terms. The first one is an \(x^2\) term, the 2nd term is an \(x\) term, and the 3rd term is a constant (merely a number). Trinomial: \(ax^2 \ + \ bx \ + \ c\)

Factoring Trinomials: a = 1

With this first example, we are going to show the way to factor a trinomial whenever \(a\), the leading coefficient is \(1\).

Example one:

Here we are factoring the trinomial: \(x^2 \ + \ 9x \ + \ 14\)

Step One: Determine values for \(b\) and \(c\). In this case, \(b \ = \ 9\) and \(c \ = \ 14\).

Step Two: Locate 2 two numbers which ADD to \(b\) and MULTIPLY to \(c\). This second step may involve a small amount of trial and error. For example, you might choose \(4\) and \(5\) since \(4 \ + \ 5 \ = \ 9\). Yet \(4 \times 5\) doesn’t come to \(28\), therefore those numbers are no good. But, if we pick \(7\) and \(2\), it is simple to confirm that:
\(7 \ + \ 2 \ = \ 9\) (\(b\)’s value); and \(7 \times 2 \ = \ 14\) (\(c\)’s value)

Step three: Utilize the numbers you chose for writing out the factors and check. In this case, the factors are \((x \ + \ 7)\) and \((x \ + \ 2)\)
Final Solution: \((x \ + \ 7)(x \ + \ 2)\)
It is possible to check the answer via multiplying the \(2\) factors (binomials) together to find out if the answer is the original trinomial like this:
\(x^2 \ + \ 9x \ + \ 14 \ = \ (x \ + \ 7)(x \ + \ 2) \ = \ x^2 \ + \ 7x \ + \ 2x \ + \ 14 \ = \ x^2 \ + \ 9x \ + \ 14\)
If you multiply the factors it equals the original trinomial.

Example two:

Factor the Trinomial: \(x^2 \ + \ 3x \ - \ 18\)

Step one: Figure out values for \(b\) and \(c\). With this example, \(b \ = \ 3\) and \(c \ = \ -18\).

Step two: Locate \(2\) numbers which ADD to \(b\) and MULTIPLY to \(c\). Locating the correct numbers will not always be as simple as it is with Example one.
In order to make factoring trinomials simpler, jot down every factor of \(c\) you can think of. With this example, \(c \ = \ -18\), therefore:
\(18 \times -1 \ = \ -18, \ 9 \times -2 \ = \ -18, \ 6 \times -3 \ = \ -18, -18 \times 1 \ = \ -18, \ ...\)
Don’t forget that the 2 numbers must multiply to \(c\) AND add to \(b\). The sole factors of \(-18\) which match both those requirements are \(6\) and \(-3\).

Step three: Utilize the chosen numbers picked to write out the factors, then and check.
The final step is then to write out the factors: \((x \ + \ 6)(x \ - \ 3)\)
\(x^2 \ + \ 3x \ - \ 18 \ = \ (x \ + \ 6)(x \ - \ 3)\)

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