How do you multiply (x + -1y)(x + y)?

2 Answers
Sep 1, 2016

x^2-y^2

Explanation:

color(blue)("Using the shortcut with a bit of explanation")

It is the case that 1y is written as just y

We also have + - . When two signs are next to each other and they are different the result is -. So + - 1y -> -y giving

(x+ - 1y)(x+y)" "->" "(x-y)(x+y)

The shortcut is to know that if you have the form a^2-b^2 then this works out to be the same as (a-b)(a+b)

That is the condition in this question so (x-y)(x+y)=x^2-y^2

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color(blue)("Demonstration that this is true")

Consider:" "color(blue)((x-y))color(brown)((x+y))

Multiply everything inside the right hand side bracket by everything inside the left hand side bracket.

color(brown)(color(blue)(x)(x+y) color(blue)(-y)(x+y)

Notice that the minus follow the color(blue)(y) in color(blue)(-y)

So we have:
color(brown)(color(blue)(x)(x+y)" " color(blue)(-y)(x+y)
x^2+xy" "-xy -y^2

x^2+0-y^2

x^2-y^2

Sep 1, 2016

x^2-y^2

Explanation:

Each term in the 2nd bracket must be multiplied by each term in the 1st bracket.

That is (color(red)(x-y))(x+y)=color(red)(x)(x+y)color(red)(-y)(x+y)

now distribute the brackets

=x^2+cancel(xy)-cancel(xy)-y^2=x^2-y^2