6.02 Multiplying polynomials
Concept summary
To multiply two polynomials together, we make use of the distributive property:
If one of the polynomials is a monomial, and the other is fully simplified, then distributing the multiplication will be the only step.
If neither polynomial is a monomial, then we apply the distributive property twice. For multiplication of two binomials, this works as follows:
| \displaystyle \left(w + x\right)\left(y + z\right) | \displaystyle = | \displaystyle \left(w + x\right)y + \left(w + x\right)z | Distribute once |
| \displaystyle = | \displaystyle y\left(w + x\right) + z\left(w + x\right) | Reorder variables | |
| \displaystyle = | \displaystyle yw + yx + zw + zx | Distribute twice |
In general, we can multiply any two polynomials using this process. Here is an example of a trinomial multiplied by a binomial:
| \displaystyle \left(x^2 + 3x + 1\right)\left(x - 2\right) | \displaystyle = | \displaystyle x\left(x^2 + 3x + 1\right) - 2\left(x^2 + 3x + 1\right) | Distribute once |
| \displaystyle = | \displaystyle x^3 + 3x^2 + x - 2x^2 - 6x - 2 | Distribute twice | |
| \displaystyle = | \displaystyle x^3 + x^2 - 5x - 2 | Combine like terms |
Notice that we can summarize the process of distributing twice as "multiply each term in the first polynomial by each term in the second polynomial, and add the results". We then simplify by combining like terms, if possible.
Worked examples
Example 1
Multiply 3 x \left( 2 x^{2} - 5 x + 4\right).
Solution
Example 2
Multiply \left( 7 y + 2\right) \left( 4 y - 5\right).