### Multiplying

When multiplying polynomials, it is critical that we correctly implement the distributive property of multiplication.  Every term in one polynomial must be multiplied by every term in the other polynomial.  It is so tempting to ignore this rule, but you are sure to have a wrong answer if you do.  Let’s look at an example.

Multiply $$\left( { - 5k - 1} \right)\left( { - 7k + 2} \right)$$. Here there are two terms in the first polynomial and two terms in the second polynomial.  Each term in the first polynomial must be multiplied by each term in the second polynomial.  There will be four terms resulting before simplification.

We have

$$\begin{gathered} \left( { - 5k - 1} \right)\left( { - 7k + 2} \right) = \left( { - 5k} \right)\left( { - 7k} \right) + \left( { - 5k} \right)\left( 2 \right) + \left( { - 1} \right)\left( { - 7k} \right) + \left( { - 1} \right)\left( 2 \right) \\ = 35{k^2} - 10k + 7k - 2 \\ \end{gathered}$$

We have our four terms.  We can combine like terms to get a final answer of $$35{k^2} - 3k - 2$$. Again, it is absolutely critical that we make sure each term in the first polynomial is multiplied by each term in the second polynomial.

When both polynomials have exactly two terms, an acronym FOIL has been created to help students remember how to multiply the polynomials.  FOIL is

F-first terms
O-outer terms
I-inner terms
L-last terms

Hopefully you can see that in the first example above, we actually did multiply the terms in the order given by FOIL.  However, FOIL is only useful when multiplying two binomials.  It would be better for us to simply understand the distributive property of multiplication.  Then we can succeed with multiplying any polynomials.  Let’s do one more example.

Multiply $$\left( {7p + 4} \right)\left( { - 8p - 3} \right)$$.

Using the distributive property of multiplication, we have

$$\begin{gathered} \left( {7p + 4} \right)\left( { - 8p - 3} \right) = \left( {7p} \right)\left( { - 8p} \right) + \left( {7p} \right)\left( { - 3} \right) + \left( 4 \right)\left( { - 8p} \right) + \left( 4 \right)\left( { - 3} \right) \\ = - 56{p^2} - 21p - 32p - 12 \\ = - 56{p^2} - 53p - 12 \\ \end{gathered}$$

5957 x

Find each product.

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4552 x

Find each product.

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4379 x

Find each product.

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