Multiplying Polynomials

By the end of this lesson, you will be able to multiply a monomial across a bracket, multiply two binomials, handle bigger products without losing terms, and predict the degree of the answer before you start.

Every Term Times Every Term

One rule covers all of it.

$$\text{multiply each term in the first by each term in the second, then gather}$$

That is the distributive property applied twice, and every named method you have heard of is a way of organizing the same multiplications.

$$(x + 3)(x + 2)$$

Four products, because two terms times two terms:

$$x \cdot x + x \cdot 2 + 3 \cdot x + 3 \cdot 2 = x^2 + 2x + 3x + 6 = x^2 + 5x + 6$$

How many products to expect. Two terms times three terms is six products. Three times three is nine. Counting them first is the cheapest possible check that you have not missed one, and missing one is the only real difficulty here.

And the degrees add. Degree $1$ times degree $1$ gives degree $2$; degree $2$ times degree $3$ gives degree $5$. So you know the answer's degree before doing any work, which is the second free check.

Multiplying by a Single Term

The simplest case, and the one everything else is built from.

$$3x^2(2x - 5) = 6x^3 - 15x^2$$

Two things happen in each product: the numbers multiply, and the powers ADD.

$$3x^2 \cdot 2x = 6x^3$$

Three times two is six, and $x^2 \cdot x^1 = x^3$. Adding the powers when multiplying is the exponent rule you met before, and it is the reason the degree of a product is the sum of the degrees.

Example 1: Two Binomials

Here is the whole method on one product, worked all the way through. $$(x + 4)(x - 3)$$

• How many products will there be? 4
• Multiply the first terms together x^2
• Multiply x by -3, and 4 by x. Write both, added -3x + 4x
• Multiply the last terms -12
• Write the whole thing gathered, in standard form x^2 + x - 12

Worked Example 1: Your Turn

$$(2x + 1)(x + 5)$$

Worked Example 2: Signs on Both

$$(3x - 2)(x - 4)$$

Bigger Products: Be Systematic

$$(x + 2)(x^2 - 3x + 5)$$

Two terms times three terms, so six products. Take the first term of the left bracket across the whole of the second, then the next.

$$x(x^2 - 3x + 5) = x^3 - 3x^2 + 5x$$

$$2(x^2 - 3x + 5) = 2x^2 - 6x + 10$$

Then add and gather:

$$x^3 - 3x^2 + 2x^2 + 5x - 6x + 10 = x^3 - x^2 - x + 10$$

The method that does not fail is one term at a time across the whole bracket, writing each line out. Trying to do all six at once is how a term goes missing, and a missing term is invisible in the answer.

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