Factor polynomials on the form of x^2 + bx + c

In previous sections we showed that when you multiply two binominals you get a trinomial.

$$\left ( x+2 \right )\left ( x-3 \right )=x^{2}+\left ( -3+2 \right )x-6=x^{2}-x-6$$

We have also shown how we can factorize a polynomial by finding a common factor among all terms. If we look at the example above we see a trinomial that does not have a common factor, but it could still be factorized by reversing the process.

We can factorize all trinomials of the form

$$x^{2}+bx+c$$

by making them into a product of two binomials

$$x^{2}+bx+c=\left ( x+m \right )\left ( x+n \right )$$

where

$$m+n=b\: \: and \: \: m\cdot n=c$$

If we use the trinomial from the example above:

$$x^{2}-x-6$$

We know that we have to find two factors of -6 whose sum is -1

It's a good idea to make a table to test all combinations

$$\begin{matrix} Factor \: of \: -6& \: \: Sum \: of \: factors\\ -1, 6& 5\\ 1, -6 & -5\\ {\color{green} 2, -3}& {\color{green} -1}\\ -2, 3 & 1 \end{matrix} $$

We see that if m = 2 and n = -3 than we will get the correct factors.

$$x^{2}-x-6=x^{2}+\left ( 2-3 \right )x+\left ( 2\cdot -3 \right )=\left ( x+2 \right )\left ( x-3 \right )$$

There is a pattern that could be good to know so that you don't have to try all different combinations of values for m and n.

If both b and c are positive then both m and n are positive.

If b is negative whereas c is positive then both m and n are negative

If c is negative then m and n do not have the same sign.

In the previous section we showed you some special patterns that could help you when you're simplifying polynomials. The same patterns can be used to help you when you're factorizing polynomials.

The difference of two squares pattern is the opposite to the sum and difference pattern

$$x^{2}-y^{2}=\left ( x+y \right )\left ( x-y \right )$$

If we use this as an example:

$$x^{2}-25=x^{2}-5^{2}=\left ( x+5 \right )\left ( x-5 \right )$$

The perfect square trinomial pattern is the opposite of the square of a binomial pattern

$$x^{2}+2xy+y^{2}=\left ( x+y \right )^{2}$$

$$x^{2}-2xy+y^{2}=\left ( x-y \right )^{2}$$

An example of this would be

$$x^{2}+14x+49=x^{2}+\left (2\cdot x\cdot 7 \right )+7^{2}=\left ( x+7 \right )^{2}$$


Video lesson

Factor the following expression

$$x^{2}+3x-10$$