Subtracting algebraic expressions

There are three steps to follow when subtracting polynomials.

1. Group like terms.

2. Subtract constant terms.

3. Subtract the coefficients of the like terms.

Note that during the subtraction:

• only like terms can be simplified;

• only the coefficients of each like term are subtracted from each other.

It is rare for an equation to only involve subtraction. If there are more than one type of operation, make sure to follow the order of operations when simplifying the algebraic expression.

Consider the following algebraic expression. ||(2x^3+3x + 2)-(x^3+2x-4)|| Brackets separate the two polynomials to be subtracted. In order to remove the brackets, the negative sign in front of the second bracket must be distributed into each of the terms inside the brackets.

The result is the following expression. ||2x^3+3x+2\color{red}{-}x^3\color{red}{-}2x\color{red}{+}4||
The expression can then be simplified like so:

1. Group the like terms (the same letters, with the same exponents). ||\color{green}{2x^3- x^3}+\color{red}{3x - 2x}+\color{blue}{2 + 4}||

2. Subtract the constant terms. ||\color{green}{2x^3- x^3}+\color{red}{3x - 2x}+\color{blue}{6}||

3. Subtract the coefficients of the like terms. ||\color{green}{x^3}+\color{red}{x}+\color{blue}{6}||

The result is |x^3+x+6.|

According to the distributive property, a negative sign in front of a bracket is distributed into each of the terms inside the brackets.

Consider the following algebraic expression. ||(5x^2-11xy+6x-13)-(-2x^2+y^2-8xy+12)||

Brackets separate the two polynomials to be subtracted. In order to remove them, the negative sign in front of the second bracket must be distributed into each of the terms inside the brackets.

The result is the following expression. ||5x^2-11xy+6x-13+2x^2-y^2+8xy-12|| The expression can be simplified as follows.

1. Group the like terms (the same letters, with the same exponents). ||\color{green}{5x^2+2x^2}\color{red}{-11xy+8xy}+6x-y^2\color{blue}{-13-12}||

2. Subtract the constant terms. ||\color{green}{5x^2+2x^2}\color{red}{-11xy+8xy}+6x-y^2\color{blue}{-25}||

3. Subtract the coefficients of the like terms. ||\color{green}{7x^2}\color{red}{-3xy}+6xy-y^2\color{blue}{-25}||

The result is |7x^2-3xy+6x-y^2-25.|

It is possible to check if the initial expression is equivalent to the simplified expression. Substitute a chosen value into each variable and calculate the value of each expression. If the two expressions have the same value, they are equivalent.

To check if the first example above is correct (|(2x^3+3x+2)-(x^3+2x-4)=x^3+x+6|), choose a value for the variables. For example, if |x=2|: ||\begin{align}2(\color{red}{2})^3+3(\color{red}{2})+2-\color{red}{2}^3-2(\color{red}{2})+4&=(\color{red}{2})^3+\color{red}{2}+6\\ \\
16+6+2-8-4+4&=8+2+6\\ \\
16&=16\end{align}||
Therefore, the two expressions are equivalent.

Consider the following two polynomials represented by algebra tiles.

|2x^3+3x+2|

|x^3+2x-4|

Note that a positive value is represented by a solid-coloured tile while a negative value is represented by a striped tile.

This is what subtracting the two polynomials looks like using algebra tiles. |(2x^3+3x+2)-(x^3+2x-4)|

After distributing the negative sign over the second polynomial, the sign of each term reverses.

Then like terms are grouped together, like in an addition.

Add the tiles. A solid-coloured tile and a striped tile cancel each other out.

The result is |x^3+x+6.|