Question

Is `a^2-b^2` equal to `(a-b)(a+b)`?

Answer 1

Yes!

This is one that is really worth committing to memory. It crops up quite often.

Explanation:

Given:

`color(blue)((a-b))color(green)( (a+b) )`

Multiply everything in the right bracket by everything in the left

`color(green)(color(blue)(a)(a+b) color(white)("dd")color(blue)(-b)(a+b))` Notice the way the minus follows the `b`

`a^2+abcolor(white)("dd")-ba+b^2`

But `-ba` has exactly the same value as `-ab` so we can write:

`a^2+cancel(ab) color(white)("dd")-cancel(ab)+b^2`

`a^2+b^2`

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Thus `a^2+b^2=(a-b)(a+b)`

Answer 2

`"yes"`

Explanation:

`a^2-b^2" is called a "color(blue)"difference of squares"`

`"since "a^2" and "b^2" are perfect squares separated by a"`
`"difference (-)"`

`"to verify this result expand the factors"`

`rArr(color(red)(a-b))(a+b)`

`=color(red)(a)(a+b)color(red)(-b)(a+b)larrcolor(blue)"distribute"`

`=a^2cancel(+ab)cancel(-ab)-b^2`

`=a^2-b^2`

`"Useful factoring tool " color(red)(bar(ul(|color(white)(2/2)color(black)(a^2-b^2=(a-b)(a+b))color(white)(2/2)|)))`