When you add polynomials, you essentially just have to combine like terms. On this page, I’ll show you a method for structuring the addition so it is easy to stay organized. If you want to do the addition in your head, you can check out the methods on this page.
How to Add Polynomials
- Fill in any missing terms and coefficients.
- Missing terms have coefficients of 0.
- Terms without coefficients have “invisible” coefficients of 1.
- Line up the like terms and add.
Examples
Add:
\[(5x^4-9x^2+3x-8)+(x^4-7x^3+5x+2)\]
Are there any missing terms or coefficients?
Yes, the first polynomial is missing an \(x^3\) term which means that \(x^3\) has a coefficient of 0.
\[5x^4{\red +0x^3}-9x^2+3x-8\]
The \(x^4\) term in the second polynomial has an “invisible” coefficient of 1. The missing \(x^2\) term has a coefficient of 0.
\[{\yellow 1}x^4-7x^3{\red +0x^2}+5x+2\]
What are the sums of the like terms?
\[ \begin{array}{lccccc} & 5 x^4 & {\red +0x^3} & -9x^2 & +3x & -8 \\ + & {\yellow 1}x^4 & -7x^3 & {\red +0x^2} & +5x & +2 \\ \hline & 6 x^4 & -7x^3 & -9x^2 & +8x & -6 \end{array}\]
The sum of \(5x^4-9x^2+3x-8\) and \(x^4-7x^3+5x+2\) is…
\[6x^4-7x^3-9x^2+8x-6\]
Add:
\[(12x^2-1)+(x^3-x^2+x)\]
Are there any missing terms or coefficients?
Yes, the first polynomial is missing an \(x^3\) term and an \(x\) term which means they have coefficients of 0.
\[{\red 0x^3}+12x^2{\red +0x}-1\]
The second polynomial is missing a constant term, which means the constant is 0. The \(x^3\), \(x^2\), and \(x\) terms have “invisible” coefficients of 1.
\[{\yellow 1}x^3-{\yellow 1}x^2+{\yellow 1}x{\red +0}\]
What are the sums of the like terms?
\[ \begin{array}{lcccc} & {\red +0x^3} & +12x^2 & {\red +0x} & -1 \\ + & {\yellow 1}x^3 & -{\yellow 1}x^2 & +{\yellow 1}x & {\red +0} \\ \hline & 1x^3 & +11x^2 & +1x & -1 \end{array} \]
The sum of \(12x^2-1\) and \(x^3-x^2+x\) is…
\[x^3+11x^2+x-1\]
I don’t have to write the coefficients of 1 in my answer, because they are implied if I write the terms with no coefficients.
Why It Works
Polynomials and integers are analogous systems. This means we can use the way we add whole numbers as an analogy to make it really easy to add polynomials. You can use the same analogy for other operations.
The terms of a polynomial are analogous to the place values of a whole number. For example, the whole number \({\yellow 4}{\green 5}{\blue 0}{\red 6}\) has 4 thousands, 5 hundreds, and 6 ones. The 0 in the tens place is a place holder because there are no tens but the place value is still important. Similarly, we can use coefficients of 0 to show the “missing” terms in polynomials.
4 Thousands, 5 Hundreds, 6 Ones
\[{\yellow 4}{\green 5}{\blue 0}{\red 6}\]
Four \(x^3\), Five \(x^2\), Six Constants
\[{\yellow 4x^3}+{\green 5x^2}+{\blue 0x}+{\red 6}\]
When we add polynomials, the process is almost identical to the way we add whole numbers. When you add whole numbers you line up the place values and add each place value. When you add polynomials you line up the like terms and add them together.
\[\begin{array}{lc|c|c|c} & 1000 & 100 & 10 & 1 \\ \hline & {\yellow 6} & {\green 2 } & {\blue 4} & {\red 3}\\ + & {\yellow 1} & {\green 3} & {\blue 5} & {\red 2}\\ \hline & {\yellow 7} & {\green 5 } & {\blue 9} & {\red 5} \end {array}\]
\[\begin{array}{lc|c|c|c} & x^3 & x^2 & x^1 & x^0 \\ \hline & {\yellow 6x^3} & {\green +2x^2 } & {\blue +4x} & {\red +3}\\ + & {\yellow 1x^3} & {\green +3x^2} & {\blue +5x} & {\red +2}\\ \hline & {\yellow 7x^3} & {\green +5x^2 } & {\blue +9x} & {\red +5} \end {array}\]
From one perspective, it is easier to add polynomials than it is to add whole numbers because you don’t have to worry about carrying numbers to the next place value.
From another perspective, it may be a little harder to add polynomials because you have to remember the rules for adding negative numbers if any of the terms have negative coefficients.
