Learning Objectives
- Find the opposite of a polynomial
- Subtract polynomials using both horizontal and vertical organization
Opposites
Find the opposite of a polynomial
When you are solving equations, it may come up that you need to subtract polynomials. This means subtracting each term of a polynomial, which requires changing the sign of each term in a polynomial. Recall that changing the sign of 3 gives [latex]−3[/latex], and changing the sign of [latex]−3[/latex] gives 3. Just as changing the sign of a number is found by multiplying the number by [latex]−1[/latex], we can change the sign of a polynomial by multiplying it by [latex]−1[/latex]. Think of this in the same way as you would the distributive property. You are distributing [latex]−1[/latex] to each term in the polynomial. Changing the sign of a polynomial is also called finding the opposite.
Example
Find the opposite of [latex]9x^{2}+10x+5[/latex].
Be careful when there are negative terms or subtractions in the polynomial already. Just remember that you are changing the sign, so if it is negative, it will become positive.Example
Find the opposite of [latex]3p^{2}–5p+7[/latex].
Notice that in finding the opposite of a polynomial, you change the sign of each term in the polynomial, then rewrite the polynomial with the new signs on each term.
Subtract polynomials
When you subtract one polynomial from another, you will first find the opposite of the polynomial being subtracted, then combine like terms. The easiest mistake to make when subtracting one polynomial from another is to forget to change the sign of EVERY term in the polynomial being subtracted.
Example
Subtract. [latex]\left(15x^{2}+12x+20\right)–\left(9x^{2}+10x+5\right)[/latex]
When polynomials include a lot of terms, it can be easy to lose track of the signs. Be careful to transfer them correctly, especially when subtracting a negative term.Example
Subtract. [latex]\left(14x^{3}+3x^{2}–5x+14\right)–\left(7x^{3}+5x^{2}–8x+10\right)[/latex]
When you have many terms, like in the example above, try the vertical approach from the previous page to keep your terms organized. However you choose to combine polynomials is up to you—the key point is to identify like terms, keep track of their signs, and be able to organize them accurately.
Example
Subtract. [latex]\left(14x^{3}+3x^{2}–5x+14\right)–\left(7x^{3}+5x^{2}–8x+10\right)[/latex]
In the following video, you will see more examples of subtracting polynomials.
Summary
We have seen that subtracting a polynomial means changing the sign of each term in the polynomial and then reorganizing all the terms to make it easier to combine those that are alike. How you organize this process is up to you, but we have shown two ways here. One method is to place the terms next to each other horizontally, putting like terms next to each other to make combining them easier. The other method was to place the polynomial being subtracted underneath the other after changing the signs of each term. In this method it is important to align like terms and use a blank space when there is no like term.