Factoring rules in algebra

Factoring is the math process of finding the factors. It can be called like finding what you have to multiply together to get an previous expression.

In this topic we review the most popular factoring methods, try to solve some example tasks and give for you best solutions.

Common factor

This method tells us that we must to find the highest possible common factor of given expression.

Factor the following expression completely: $3x^2-6x$

$$3x^2-6x=3x(x-2)$$

Note! $3x^2-6x=x(3x-6)$ or $3x^2-6x=3(x^2-2x)$ is not solutions, because the factor is not the highest.

Factoring with formulas

Here is a list of common factoring formulas (Short multiplication formulas) that can makes factoring much easier.

\begin{align} \displaystyle a^2 + 2ab+b^2 &=(a + b)^2 \\ \displaystyle a^2 - 2ab+b^2 &=(a - b)^2 \\ \displaystyle a^2-b^2 &=(a+b)(a-b) \\ \displaystyle a^3 + 3a^2b+3ab^2 + b^3 &=(a + b)^3 \\ \displaystyle a^3 - 3a^2b+3ab^2 - b^3 &=(a - b)^3 \\ \displaystyle a^3 - b^3 &=(a - b)(a^2 + ab+b^2) \\ \displaystyle a^3 + b^3 &=(a + b)(a^2 - ab+b^2) \end{align}

Factor the following expression completely: $x^3-x$

$$x^3-x=x(x^2-1)=x(x-1)(x+1)$$

Factor the following expression completely: $28x^2-28x+7$

$$28x^2-28x+7=7(4x^2-4x+1)\\=7(2x-1)^2$$

Quadratic function

Sometimes, when we have quadratic function, the factoring formulas above won't work. If it is, we can equal this function to zero and solve it with discriminant or other method. The solutions $x_1$ and $x_2$ we got can be used in formula below.

$$ax^2+bx+c=a(x-x_1)(x-x_2)$$

Factor the following expression completely: $3x^2-3x-18$

$$3x^2-3x-18=0\\x_1=-2 \ \text{and} \ x_2=3\\3x^2-3x-18=3(x+2)(x-3)$$

Factoring by grouping

If we have expression that have four terms, we usually should group the first two terms together and the last two terms together. Then we can try to find common factors of both groups. After all we should get common factor of both groups.

Factor the following expression completely: $x^3-3x^2+4x-12$

$$x^3-3x^2+4x-12=\underbrace{x^3-3x^2}+\underbrace{4x-12}\\ =x^2(x-3)+4(x-3)=(x-3)(x^2+4)$$

Note! Not all the time the best way is to group the first two terms together and the last two terms together. Sometimes we should group in other order.

2020-12-01