Polynomial Operations Calculator (2024)

Polynomial Calculators

• FactoringPolynomials

• Polynomial Roots
• Synthetic Division
• PolynomialOperations
• GraphingPolynomials
• Simplify Polynomials
• Generate From Roots

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• Multiplication / Division
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Radical Expressions

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• Simplifying

Solving Equations

• Quadratic Equations Solver

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• Solving Equations - WithSteps

Quadratic Equation

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• Quadratic Plotter
• Factoring Trinomials

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• Distance calculator

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  See Also

  GCSE Exam Boards Explained: Which is Hardest? - Achieve LearningCambridge International Mathematics IGCSE 0580/43 Ppaer 4 ... - [PDF Document]1.22: Surface Area of Common Solids

• Circle From 3 Points
• Circle-line Intersection

Complex Numbers

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Systems of equations

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Matrices

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Calculus Calculators

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Sequences & Series

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Trigonometry

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Numbers

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Financial Calculators

• Simple Interest

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Other Calculators

• Sets

• Work Problems

$(3x-4)+(5+3x-4x^2)$

$(9x+5)-(3x-3)$

$(-2x^6 + x^5 - 3x^2 - 4x + 7) - (x^5 + 2x^2 - 4x + 4)$

$(2x+3)\cdot(5x-3)$

$(x^2 - x + 3)\cdot(2x^2 + 4x - 3)$

$\dfrac{x^3 + x^2 + 4}{x + 2}$

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 binomialmultiplication 4. standard multiplication 5. division by monomialand 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 frontof 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 theother.

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

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

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

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 secondbracket.

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

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 anexample:

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

$$ \begin{array}{lcccc}\text{First} & : & 5a & \cdot & 2a & = & 10a^2 \\\text{Outer} & : & 5a & \cdot & -3 & = & -15a \\\text{Inner} & : & 2 & \cdot & 2a & = & 4a \\\text{Last} & : & 2 & \cdot & -3 & = & -6 \\\hline\end{array} $$
$$\begin{aligned}(5a + 2) \cdot(2a - 3) &= \underbrace{10a^2}_{F} - \underbrace{15a}_{O} + \underbrace{4a}_{I} -\underbrace{6}_{L} = \\[1em]&= 10a^2 - 11a - 6\end{aligned}$$