How to Add and Subtract Polynomials
- Identify the “invisible” coefficients of 1.
- Distribute the coefficients outside of the parentheses to all the terms inside parentheses.
- Combine like terms.
\[(3x^4-2x^3+7x-1)+(x^3-6x^2+8x-5)\]
What are the “invisible” coefficients outside the parentheses that need to be distributed to the terms inside?
The first polynomial has an invisible coefficient of +1 outside the parentheses. The second polynomial also has a +1 coefficient because we are adding the second polynomial to the first.
\[{\green+1}(3x^4-2x^3+7x-1){\purple +1}(x^3-6x^2+8x-5)\]
When I multiply the terms inside the parentheses by +1, all of the coefficients stay the same.
\[{\green +3}x^4{\green -2}x^3{\green+7}x{\green -1}{\purple +1}x^3{\purple -6}x^2{\purple +8}x{\purple -5}\]
Are there any like terms?
Yes, there are two \(\red -x^3\) terms, two \(\yellow x\) terms, and two constant terms.
\[3x^4{\red -2x^3}{\yellow +7x}{\blue -1}{\red +1x^3}-6x^2{\yellow +8x}{\blue -5}\]
When I combine these like terms, the simplified expression is…
\[3x^4{\red -1x^3}-6x^2{\yellow +15x}{\blue -6}\]
The sum of \(3x^4-2x^3+7x-1\) and \(x^3-6x^2+8x-5\) is…
\[3x^4-x^3-6x^2+15x-6\]
