Multiplying polynomials

Multiplying $(x + 3) (x + 5)$ is just the distributive property twice. Picture a rectangle with sides $x + 3$ and $x + 5$ , split into four smaller rectangles.

	$x$	$+ 5$
$x$	$x^{2}$	$5 x$
$+ 3$	$3 x$	$15$

Work through it

$(x + 3) (x + 5)$

Multiply every pair. $x^{2} + 5 x + 3 x + 15$

Combine the middle terms. $x^{2} + 8 x + 15$

$(2 x - 1) (x + 4)$

Multiply every pair. Keep the minus with the 1. $2 x^{2} + 8 x - x - 4$

Combine. $2 x^{2} + 7 x - 4$

Special products

Two patterns come up so often they're worth seeing:

(a + b)^{2} = a^{2} + 2 ab + b^{2} (a + b) (a - b) = a^{2} - b^{2}

$(x - 6)^{2}$

This means multiply it by itself. $(x - 6) (x - 6)$

Use the pattern, including the middle term. $x^{2} - 12 x + 36$

Watch out for this

$(x + 3)^{2} = x^{2} + 9$ .

You've dropped the middle term. Test $x = 1$ : $(1 + 3)^{2} = 16$ , but $1 + 9 = 10$ .

The short version
Multiply each term in the first by each term in the second, then combine like terms.

Your turn

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