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- Polynomial Factoring Calculator

This calculator is a free online math tool that writes a polynomial in factored form.

The solver shows a complete step-by-step explanation.

**problem**

Factor polynomial $ 5ab+2b+5ac+2c $.

**solution**

**explanation**

To factor $ 5ab+2b+5ac+2c $ we can use **factoring by grouping**.

Group $ \color{blue}{ 5ab }$ with $ \color{blue}{ 2b }$ and $ \color{red}{ 5ac }$ with $ \color{red}{ 2c }$ then factor each group.

$$ \begin{aligned} 5ab+2b+5ac+2c &= ( \color{blue}{ 5ab+2b } ) + ( \color{red}{ 5ac+2c }) = \\ &= \color{blue}{ b( 5a+2 )} + \color{red}{ c( 5a+2 ) } = \\ &= (b+c)(5a+2) \end{aligned} $$
working...

EXAMPLES

example 1:ex 1:

factor $x^2 - 10x + 25$

example 2:ex 2:

factor $3x^2 - 27$

example 3:ex 3:

factor $4x^3 - 21x^2 + 29x -6$

example 4:ex 4:

factor $3uv-12u+6v-24 $

example 5:ex 5:

factor $a^2 + 2a + 1 - b^2$

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TUTORIAL

To find the factored form of a polynomial, this calculator employs the following methods:

**1**. Factoring GCF, **2** Factoring by grouping, **3** Using the difference of squares, and **4** Factoring Quadratic Polynomials

**Example 01:** Factor $ 3ab^3 - 6a^2b $

The method is very useful for finding the factored form of the four term polynomials.

**Example 03:** Factor $ 2a - 4b + a^2 - 2ab $

We usually group the first two and the last two terms.

$$ 2a - 4b + a^2 - 2ab = \color{blue}{2a - 4b} + \color{red}{a^2 - 2ab} $$We now factor $ \color{blue}{2} $ out of the blue terms and $ \color{red}{a} $ out of from red ones.

$$ \color{blue}{2a - 4b} + \color{red}{a^2 - 2ab} = \color{blue}{2 \cdot (a - 2b)} + \color{red}{a \cdot (a - 2b) } $$At the end we factor out common factor of $ (a - 2b) $

$$ 2 \cdot \color{blue}{(a - 2b)} + a \cdot \color{blue}{(a - 2b)} = (a - 2b) (2 + a) $$**Example 04:** Factor $ 5ab + 2b + 5ac + 2c $

This is a rare situation where the first two terms of a polynomial do not have a common factor, so we have to group the first and third terms together.

$$ \begin{aligned} 5ab + 2b + 5ac + 2c &= \color{blue}{5ab + 5ac} + \color{red}{2b + 2c} = \\ &=\color{blue}{5a (b + c)} + \color{red}{2 (b + c)} = \\ &= (b+c)(\color{blue}{5a} + \color{red}{2}) \end{aligned} $$The most common special case is the **difference of two squares**

We usually use this method when the polynomial has only two terms.

**Example 05:** Factor $ 4x^2 - y^2 $.

First, we need to notice that the polynomial can be written as the difference of two perfect squares.

$$ \color{blue}{4x^2} - \color{red}{y^2} = \color{blue}{ \left(2x\right)}^2 - \color{red}{y}^2 $$Now we can apply above formula with $ \color{blue}{a = 2x} $ and $ \color{red}{b = y} $

$$ \left( \color{blue}{2x} \right)^2 - \color{red}{y}^2 = (2x - b) (2x + b) $$**Example 06:** Factor $ 9a^2b^4 - 4c^2 $.

The binomial we have here is the difference of two perfect squares, thus the calculation will be similar to the last one.

$$ 9a^2b^4 - 4c^2 = \left( 3ab^2 \right)^2 - \left( 2c \right)^2 = (3ab^2-2c)(3ab^2+2c) $$The second special case of factoring is the **Perfect Square Trinomial**

**Example 07:** Factor $ 100 + 20x + x^2 $

In this case, the first and third terms are perfect squares. So we can use the above formula.

$$ \color{blue}{100} + 20x + \color{red}{x^2} = \color{blue}{10}^2 + 2 \cdot \color{blue}{10} \cdot \color{red}{x} + \color{red}{x}^2 = (10 + x)^2 $$The best way to explain this method is by using an example.

**Example 08:** Factor $ x^2 + 3x + 4 $

We know that the factored form has the following pattern

$$ x^2 + 5x + 4 = (x + \_ ) (x + \_ ) $$All we have to do now is fill in the blanks with the two numbers. To do this, notice that the product of these two numbers has to be 4 and their sum has to be 5. We conclude, after some trial and error, that the missing numbers are $ \color{blue}{1} $ and $ \color{red}{4} $. As a result, the solution is::

$$ x^2 + 5x + 4 = (x + \color{blue}{1} ) (x + \color{red}{4} ) $$**Example 09:** Factor $ x^2 - 8x + 15 $

Like in the previous example, we look again for the solution in the form

$$ x^2 - 8x + 15 = (x + \_ ) (x + \_ ) $$ The sum of missing numbers is $-8$ so we need to find two **negative** numbers such that the product is $15$ and the sum is $-8$.
It's not hard to see that two numbers with such properties are $-3$ and $-5$, so the solution is.

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