Calculator for polynomial operations

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) $

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