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Subtracting Polynomials

First, let's subtract the coefficients of terms with the same exponents.

Be careful when distributing the "-" sign through each polynomial in the equation. Recall the rules for multiplying negative numbers!

Let's look at an example:

(Video) Adding and Subtracting Polynomials

by PatrickJMT

This video by PatrickJMT shows how you subtract and add polynomials.

$\large{\text{Summary}}$

First, we need to combine like terms. Let's look at an example to see this. Suppose we want to subtract: $$2(x^4+5x)-6(x^4+8x-3)$$

First, we distribute anything outside the parentheses inside. It's important to note that when subtracting polynomials, we need to distribute the negative sign inside as well. In our example, we distribute the $2$ inside, and then the $-6$ inside. Once this is done, we can drop the parentheses. $$\begin{align} &2(x^4+5x)-6(x^4+8x-3)\\ &=2x^4+10x{\color{#8925AE}{-}}6x^4{\color{#8925AE}{-}}48x{\color{#8925AE}{+}}18 \end{align}$$

Next, we want to group the like terms together. Recall, like terms have the same variable and same exponent. $$\begin{align} &2x^4+10x-6x^4-48x+18\\ &=(2x^4-6x^4)+(10x-48x)+18 \end{align}$$

Then, pull out the variables. For example, in $(10x-48x)$, both have an $x$ so we can pull it out. $$\begin{align} &(2x^4-6x^4)+(10x-48x)+18\\ &=(2-6)x^4+(10-48)x+18 \end{align}$$

Finally, subtract the coefficients. $$\begin{align} &(2-6)x^4+(10-48)x+18\\ &=-4x^4-38x+18 \end{align}$$

Now that you've learned how to subtract single-variable polynomials, you can learn to subtract multi-variable polynomials!

While solving equations, it often comes in handy to be able to subtract polynomials. Let us begin with an example: $$8x^3+54x^2-23x+4=4x^3-6x^2+19x-17$$ We first begin by moving all the terms to one side of the equation: $$8x^3+54x^2-23x+4-4x^3+6x^2-19x+17=0$$ Now we put them into groups based on the exponent of $x$. $$(8x^3-4x^3)+(54x^2+6x^2)+(-23x-19x)+(4+17)=0$$ By subtracting/adding, we get: $$4x^3+60x^2+(-42x)+21=0$$ $$4x^3+60x^2-42x+21=0$$ This is the final answer.

Let's do another example: $$3y^2+x=-2x-8$$ Looking at this, we see that there are no other $y^2$ terms in this equation besides $3y^2$, so we can leave it alone. However, we notice that there are $2$ $x$ terms, so we can combine them. We move the $-2x$ to the other side of the equation and combine: $$3y^2+x+2x+8=0$$ $$3y^2+3x+8=0$$ Note that we also moved the $-8$ to the other side of the equation. It is always a good idea to set the whole equation equal to $0$, as it will help when performing other operations on polynomials.