Factoring Trinomials of the Form \(x^2+bx+c\)

Hello! Welcome to this video on factoring trinomials of the form \(x^{2}+bx+c\). An example of a trinomial like this is \(x^{2}+8x+15\). Let’s go ahead and factor this trinomial. When we factor a trinomial, we want to break the trinomial into two expressions that, when multiplied together, gives you the original trinomial. Each of these expressions is a factor of the original trinomial, and they can help you find solutions to equations.

When we factor a trinomial, we are going to use a process that is the reverse of the FOIL method. We start by writing two sets of parentheses.

\(( )( )\)

Now, we consider the first term, \(x^{2}\). What two things multiply together to get \(x^{2}\)? \(x\cdot x=x^{2}\), so \(x\) will be the first term in both sets of parentheses.

\((x )(x )\)

Then, we need to find two numbers that multiply to the third term, 15, and add to the coefficient of the second term, 8. \(3\cdot 5=15\) and \(3+5=8\), so our two numbers will be 3 and 5. Write \(+3\) in one set of parentheses after the \(x\) and \(+5\) in the other set of parentheses.

\((x+3)(x+5)\)

Remember, factoring a trinomial is like “un-FOILing” it, so if we apply the FOIL method to this new expression, we should get our original trinomial. Let’s do that now to check our work.

\(x^{2}+5x+3x+15\)
 
\(x^{2}+8x+15\)

Sure enough, we got our original trinomial, which means our factored form of \((x+3)(x+5)\) is correct!

Let’s try another one. Factor the trinomial \(x^{2}+2x-63\).

First, set up two sets of parentheses.

\(( )( )\)

Then, determine what two things multiply to \(x^{2}\). If the coefficient of \(x^{2}\) is 1, like all the examples in this video, it will always be \(x\cdot x\), so let’s go ahead and write an \(x\) in each set of parentheses.

\((x )(x )\)

Now, we need to find two numbers that multiply to \(–63\) and add to 2. Let’s consider the factors of 63. We have 1 and 63, 3 and 21, and 7 and 9. Now, since our product is negative, we know that one of the factors will be positive and one will be negative. So we need to consider which two factors (remember, one being negative and one being positive) will add to 2. The two factors must be \(–7\) and 9. Write \(-7\) in one set of parentheses and \(+9\) in the other.

\((x-7)(x+9)\)

If we FOIL this out, we will get \(x^{2}+2x-63\), which is our original trinomial, so our factorization is correct.

Before we go, let’s work through one more example. Factor the trinomial \(x^{2}-13x+22\).

First, set up the parentheses.

\(( )( )\)

Then, put the \(x\)’s where they belong, since \(x\cdot x=x^{2}\).

\((x )(x )\)

Finally, we need to find two numbers that multiply to 22 and add to \(–13\). Since the two numbers add to a negative number but multiply to a positive number, both numbers have to be negative.

Let’s consider the factors of 22. We have \(1\times 22\), and \(2\times 11\). Which set of factors, when both numbers are negative, add to \(–13\)? The only option is \(–2\) and \(–11\), so we need to write \(-2\) in one set of parentheses and \(-11\) in the other set.

\((x-2)(x-11)\)

We can FOIL this out and get \(x^{2}-13x+22\), so that means our factored form, \((x-2)(x-11)\) is correct.

Now there is one thing to note before we go. In the examples in this video, our trinomials factored out nicely. This isn’t possible for every single trinomial. And when you come across things like that, there are other options like completing the square or using the quadratic formula to solve. However, when you are asked to factor trinomials of this form, you’ll know how to do it.

I hope that this video on factoring trinomials of the form x2+bx+c was helpful. Thanks for watching, and happy studying!