# When multiplying or dividing in algebra with two different signs (plus or minus sign) how do you know which sign to put in the final answer?

Example: Multiplying (2r + 2)(2r - 2)

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• What you've done is solve for the numbers that will make the equation true. In a quadratic equation there are two numbers that will do that. So, the answer includes both answers.

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• If you are simplifying this, you'd want to put it into standard polynomial form.  The highest exponents are on the left down to the constants on the right.

This is the factored form of the difference of two squares, so simplifying it would put you back to that difference of two squares:

4r² - 4

If the question is which one to put them in if you need to factor:

(2r + 2)(2r - 2) or (2r - 2)(2r + 2)

They are essentially the same answer and any math teacher with any merit will count both as correct.

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• (2r + 2)(2r - 2)

= 4r^2 - 4

= 4(r + 1)(r - 1)

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• First, you remember arithmetic. Multiplying or dividing a positive number by a negative number produces a negative result. Multiplying or dividing a negative number by a negative number produces a positive result.

For example, (-6) / (-3) = 2

The same remains true of coefficients in algebra.

(2r)(-2) = -4r

When you multiply out (1x + 2)(3x - 4), you get:

(1x)(3x) + (1x)(-4) + (2)(3x) + (2)(-4)

= 3x^2 + (-4x) + 6x + (-8)

= 3x^2 + 2x - 8

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• ( 2r  +  2 ) ( 2r -  2 )

4r²  - 4r + 4r - 4

4r² - 4

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• Use FOIL for starters with binomials.  Here's a handy little trick when multiplying or dividing with opposite signs.  Count the number of -ve signs.  If you have an odd number of -ve signs the answer is -ve. An even number of -ve signs the answer is +ve.

In your example you have one negative sign so the number will be -ve.

+2 X -2 = -4

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• very simple, term by term

+ multiplied by + equals +

– multiplied by – equals +

+ multiplied by – equals –

– multiplied by + equals –

even simpler rule:

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•  (2r + 2)(2r - 2)

= 4r^2 - 4

= 4(r + 1)(r - 1)

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• You are multiplying two binomials. You can easily use the FOIL method.

FOIL = First, Outer, Inner, Last

First --> 2r * 2r = 4r²

Outer --> 2r * -2 = -4r

Inner --> 2 * 2r = 4r

Last --> 2 * -2 = -4

4r² - 4r + 4r - 4

The r terms in the middle cancel to 0.

4r² - 4

If you are using the shortcut rule (a + b)(a - b), that's a *difference* of square = a² - b². It's pretty clear the final result should be a difference if you look at the last term. You have +b and -b so the result will be negative --> -b²

(2r + 2)(2r - 2)

a = 2r

b = 2

a² - b²

= (2r)² - 2²

= 4r² - 4

Summary:

An even number of negatives cancel out (-2 * -2 * -2 * -2 = 16)

If you have an *odd* number of negatives, they stay negative (-2 * -2 * -2 = -8)

So in your case with 2 * -2, you have an *odd* number of negatives and the result stays negative.

2 * -2 = -4

Because positive * negative = negative.

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• Term by term:

plus times plus = plusminus times minus = plusplus time minus = minusExample:(2r + 2)(2r - 2)= 2r(2r - 2) + 2(2r-2)= (2r)(2r) + (2r)(-2) + 2(2r) + 2(-2)= (4r²) + (-4r) + (4r) + (-4)[2r*2r is plus times plus = plus 4r² 2r times -2 is plus times minus = minus 4r 2 times 2r = plus times plus = plus 4r 2 times -2 = plus times minus = minus 4]= 4r² - 4r + 4r - 4= 4r² - 4

= 4(r²-1)

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