Operations on polynomials calculator

This solver performs basic arithmetic operations on polynomials (addition, subtraction, multiplication and division).
Calculator shows complete work and detailed step-by-step explanation for each operation.

Operations on polynomials - with steps
add, subtract, multiply and divide polynomials
show help ↓↓ examples ↓↓ tutorial ↓↓
(3x^2+4x-3) - (8x+5)
(3x-5)(x^2+2x-4)
Add polynomials Subtract polynomials
Multiply polynomials (default) Divide polynomials
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examples
example 1:ex 1:
$(3x-4)+(5+3x-4x^2)$
example 2:ex 2:
$(9x+5)-(3x-3)$
example 3:ex 3:
$(-2x^6 + x^5 - 3x^2 - 4x + 7) - (x^5 + 2x^2 - 4x + 4)$
example 4:ex 4:
$(2x+3)\cdot(5x-3)$
example 5:ex 5:
$(x^2 - x + 3)\cdot(2x^2 + 4x - 3)$
example 6:ex 6:
$\dfrac{x^3 + x^2 + 4}{x + 2}$
TUTORIAL

Operations on polynomials

In this short tutorial you will learn how to perform basic operations on polynomial.
The basic operations are 1. addition 2. subtraction 3. FOIL method for binomial multiplication 4. standard multiplication 5. division by monomial and 6. long division. Note that this calculator displays a step-by-step explanation for each of these operations. Nevertheless, let's start with addition.

1A: Polynomial addition - horizontal

Example 01: Add $ (2a+5) + (4a-3) $

First we will remove parenthesis because there are no minus sign in front of the bracket:

$$ (2a+5) + (4a-3) = 2a + 5 + 4a - 3 $$

Now we will group the like terms:

$$ 2a + 5 + 4a - 3 = 2a + 4a + 5 - 3$$

At the end we combine like terms:

$$ 2a + 4a + 5 - 3 = 6a + 2 $$

Putting all together we have

$$ \begin{aligned} (2a+5) + (4a-3) \overbrace{=}^{\text{remove par.}}& \color{blue}{2a} + 5 + \color{blue}{4a} - 3 = \\ \overbrace{=}^{\text{group like terms}}& \color{blue}{2a + 4a} + 5 - 3 = \\ \overbrace{=}^{\text{combine like terms}}& \color{blue}{6a} + 2 \end{aligned} $$

1B: Polynomial addition - vertical

Example 02: Add $ (5x^3 - 3x^2 - 2x + 5) + (-x^3 + 2x^2 - 7) $

To perform vertical addition we need put like terms one below the other.

$$ \begin{array}{rrr} \color{blue}{5x^3} & \color{orangered}{-3x^2} & -2x & \color{purple}{5} \\ \color{blue}{-x^3} & \color{orangered}{2x^2} & & \color{purple}{-7} \\ \hline \end{array} $$

Now it is very easy to combine like terms

$$ \begin{array}{rrr} \color{blue}{5x^3} & \color{orangered}{-3x^2} & -2x & \color{purple}{5} \\ \color{blue}{-x^3} & \color{orangered}{2x^2} & & \color{purple}{-7} \\ \hline \color{blue}{4x^3} & \color{orangered}{-x^2} & -2x & \color{purple}{-2} \end{array} $$

The answer is

$$ 4x^2 -x^2-2x-2 $$

2B: Polynomial subtraction - vertical

Example 04: Subtract $ (2x^2 + x - 3) - (x^2 - 3x + 5) $

Put like terms one below the other and but we have to change all the signs in the second polynomial.

$$ \begin{array}{rrrr} 2x^2 & x & -2 \\ \color{red}{\bf{-}}x^2 & \color{red}{\bf {+}}3x & \color{red}{\bf{-}}5 \\ \hline \end{array} $$

Now we can combine like terms

$$ \begin{array}{rrrr} 2x^2 & x & -2 \\ -x^2 & 3x & -5 \\ \hline x^2 & 4x & -7 \\ \end{array} $$

So the answer is $ x^2 + 4x - 7 $

2A: Polynomial subtraction - horizontal

Example 03: Subtract $ (5x - 7) - (3x - 3) $

Here we remove parenthesis by change the sign of every term in second bracket

$$ (5x - 7) \color{blue}{- (3x - 3)} = 5x - 7 \color{blue}{- 3x + 3} $$

Now, as in previous example, group the like terms ...

$$ 5x - 7 -3x + 3 = 5x - 3x -7 + 3 $$

...and combine like terms:

$$ 5x - 3x - 7 + 3 = 2x - 4 $$

Putting all together we have

$$ \begin{aligned} (5x-7) - (3x-3) \overbrace{=}^{\text{remove par.}}& 5x - 7 - 3x + 3 = \\ \overbrace{=}^{\text{group like terms}}& 5x - 3x - 7 + 3 = \\ \overbrace{=}^{\text{combine like terms}}& 2x - 4 \end{aligned} $$

3: Polynomial multiplication - FOIL

We use this method for multiplying two binomials. FOIL method can be best described by using an example:

Example 05: Use FOIL method to multiply $ (5a + 2) \cdot(2a - 3) $

$$ \begin{array}{lcccc} \text{First} & : & 5a & \cdot & 2a & = & 10a^2 \\ \text{Outer} & : & 5a & \cdot & -3 & = & -15a \\ \text{Inner} & : & 2 & \cdot & 2a & = & 4a \\ \text{Last} & : & 2 & \cdot & -3 & = & -6 \\ \hline \end{array} $$

Now we combine terms at the right

$$ (5a + 2) \cdot(2a - 3) = \underbrace{10a^2}_{F} - \underbrace{15a}_{O} + \underbrace{4a}_{I} - \underbrace{6}_{L} = 10a^2 - 11a - 6 $$

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