The multiplication algorithm is my favorite way to multiply polynomials because it’s just the polynomial version of this familiar method we use to multiply regular numbers.

The multiplication algorithm is one of four methods you can use to multiply polynomials. I recommend that you learn all four methods and then choose the method that makes the most sense to you.

The other three methods are:

## How to Use the Multiplication Algorithm to Multiply Polynomials

When you multiply polynomials with the standard multiplication algorithm, the steps are very similar to the steps you use to multiply normal numbers.

- Line up the terms of each polynomial by “place value”.
- Multiply each term in the second polynomial by
**ALL**the terms in the first polynomial. - Combine like terms by adding up each “place value” column.

\[ \begin {array}{ccc} & {\red 2} & {\yellow 3} \\ \times & {\blue 1} & {\green 2} \\ \hline & {\green 4} & {\green 6} \\ {\blue 2} & {\blue 3} & \\ \hline 2 & 7 & 6 \end {array} \]

\[ \begin {array}{ccc} & {\red 6x} & {\yellow -3} \\ \times & {\blue 5x} & {\green +2} \\ \hline & {\green 12x} & {\green -6} \\ {\blue 30x^2} & {\blue -15x} & \\ \hline 30x^2 & -3x & -6 \end {array} \]

Normal Numbers

\[{\red 2}{\yellow 3}\times{\blue 1}{\green 2}\]

\[ \begin {array}{ccc} & {\red 2} & {\yellow 3} \\ \times & {\blue 1} & {\green 2} \\ \hline & {\green 4} & {\green 6} \\ {\blue 2} & {\blue 3} & \\ \hline 2 & 7 & 6 \end {array} \]

Polynomials

\[({\red 6x}{\yellow -3})\times({\blue 5x}{\green +2})\]

\[ \begin {array}{ccc} & {\red 6x} & {\yellow -3} \\ \times & {\blue 5x} & {\green +2} \\ \hline & {\green 12x} & {\green -6} \\ {\blue 30x^2} & {\blue -15x} & \\ \hline 30x^2 & -3x & -6 \end {array} \]

Notice the similarities and differences between the process for multiplying normal numbers and the process for multiplying polynomials.

The multiplication algorithm is almost identical EXCEPT for the fact that the polynomials have monomials with increasing degrees (\(x^0\), \(x^1\), \(x^2\), \(x^3\), etc.) instead of normal place values.

- Normal Place Values: 1, 10, 100, 1000, etc.
- Polynomial Place Values: (\(1\), \(x\), \(x^2\), \(x^3\), etc.).

I usually simplify the first two place values because \(x^0=1\) and \(x^1=x\) regardless of what x is.

If you have any negative coefficients, it’s important to follow the rules for multiplying negative numbers when you multiply the terms of the polynomials.

## Example

Multiply:

\[(3x^4-7x^3+2x-5)(4x^2-6x+9)\]

I’ll start by labeling the place value columns and then write the terms of each polynomial in the correct columns.

The first polynomial is missing an \(x^2\) term so I’m going to write \(0x^2\) in that column as a place holder.

\[\begin{array}{c|c|c|c|c|c|c} x^6 & x^5 & x^4 & x^3 & x^2 & x & 1 \\ \hline & & {\blue 3x^4} & {\blue -7x^3} & {\purple 0x^2} & {\blue +2x} & {\blue -5}\\ & & & & {\red 4x^2} & {\yellow -6x} & {\green +9} \\ \hline & & & & & & \end{array}\]

Then, I’ll multiply EVERY term in the first polynomial by the 9 in the second polynomial.

Next, I’ll multiply EVERY term in the first polynomial by the -6x in the second polynomial.

Finally, I’ll multiply EVERY term in the first polynomial by the \(4x^2\) in the second polynomial.

Lastly, I’ll combine like terms by adding up all the terms in each place value column.

Answer:

\((3x^4-7x^3+2x-5)(4x^2-6x+9)\)

\(=\)

\(12x^6-46x^5+69x^4-55x^3-32x^2+48x-45\)

## Why It Works

When you’re multiplying polynomials, you have to make sure that each term in the first polynomial is multiplied by each term in the second polynomial.

The multiplication algorithm is a great way to organize this process because it’s so similar to the multiplication method you memorized in elementary school.

When one of the terms in the first polynomial is multiplied by all of the terms in the second polynomial, that multiplication is recorded in one of the rows between the two black lines.

When the multiplication is organized like this, it’s easy to catch your mistake if you accidentally skip a term.

It’s also nice because the like terms are automatically lined up in the place value columns. This makes it really easy to add them together when it’s time to combine like terms.

The best part is you don’t even have worry about “carrying” between the place values like you do with normal multiplication.