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Calculator for polynomial operations
This solver performs arithmetic operations on polynomial, including addition, subtraction, multiplication and division. For multiplication, it uses both the GRID and FOIL methods. The calculator displays complete work as well as a detailed step-by-step explanation for each operation.
Multiply polynomial $ \, \color{blue}{ 5a+2 } \, $ by $ \, \color{orangered}{ 2a-3 }$.
solution
$$ \left( \color{blue}{5a+2} \right) \cdot \left( \color{orangered}{2a-3} \right) = 10a^2-11a-6 $$explanation
In this example we are multiplying two binomials so FOIL method can be used.
$$ \begin{aligned} \left( \color{blue}{ 5a+2}\right) \cdot \left( \color{orangered}{ 2a-3}\right) &= \underbrace{ \color{blue}{5a} \cdot \color{orangered}{2a} }_{\text{FIRST}} + \underbrace{ \color{blue}{5a} \cdot \left( \color{orangered}{-3} \right) }_{\text{OUTER}} + \underbrace{ \color{blue}{2} \cdot \color{orangered}{2a} }_{\text{INNER}} + \underbrace{ \color{blue}{2} \cdot \left( \color{orangered}{-3} \right) }_{\text{LAST}} = \\ &= 10a^2 + \left( -15a\right) + 4a + \left( -6\right) = \\ &= 10a^2 + \left( -15a\right) + 4a + \left( -6\right) = \\ &= 10a^2-11a-6; \end{aligned} $$Operations on polynomials
In this short tutorial, you will learn how to perform basic operations on polynomials.
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 the parenthesis because there are no minus sign in front of the brackets:
$$ (2a+5) + (4a-3) = 2a + 5 + 4a - 3 $$We'll now group the like terms:
$$ 2a + 5 + 4a - 3 = 2a + 4a + 5 - 3$$Finally, 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 must arrange like terms one above the other.
$$ \begin{array}{rrr} \color{blue}{5x^3} & \color{orangered}{-3x^2} & -2x & \color{purple}{5} \\ \color{blue}{-x^3} & \color{orangered}{2x^2} & & \color{purple}{-7} \\ \hline \end{array} $$It is now quite simple 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) $
Place like terms one above the other, but in the second polynomial, we must now alter all of the signs.
$$ \begin{array}{rrrr} 2x^2 & x & -2 \\ \color{red}{\bf{-}}x^2 & \color{red}{\bf {+}}3x & \color{red}{\bf{-}}5 \\ \hline \end{array} $$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 changing the sign of every term in the 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 them:
$$ 5x - 3x - 7 + 3 = 2x - 4 $$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
This method is used to multiply two binomials. The best way to explain the FOIL method is to use an example:
Example 05: Use FOIL method to multiply $ (5a + 2) \cdot(2a - 3) $
$$ \begin{array}{lcccc} \text{First} & : & 5a & \cdot & 2a & = & 10a^2 \\ \text{Outer} & : & 5a & \cdot & -3 & = & -15a \\ \text{Inner} & : & 2 & \cdot & 2a & = & 4a \\ \text{Last} & : & 2 & \cdot & -3 & = & -6 \\ \hline \end{array} $$$$ \begin{aligned} (5a + 2) \cdot(2a - 3) &= \underbrace{10a^2}_{F} - \underbrace{15a}_{O} + \underbrace{4a}_{I} - \underbrace{6}_{L} = \\[1em] &= 10a^2 - 11a - 6 \end{aligned} $$
1. Operations with Polynomials — with step-by-step examples.
2. Video tutorial — on how to multiply polynomials.
3. Polynomial long division — calculator with steps