# Extending the Distributive Property. You already know the Distributive Property …

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Extending the Distributive Property

You already know the Distributive Property

So far you have used it in problems like this:

The distributive property is used all the time with polynomials.

One thing it lets us do is multiply a monomial times a larger polynomial.

You normally wouldnt show the work, but this is the distributive property.

You just take the monomial times each of the terms of the polynomial, one at a time.

So 3x2(5x2 2x + 3)

= 15x4 6x3 + 9x2When you multiply each term, its the basic rules of multiplying monomials.

Multiply the coefficients.Add the exponents.Multiply:

2n5(3n3 + 5n2 8n 3)

8x4y3(2x2y2 + 7x5y)

Multiply:

2n5(3n3 + 5n2 8n 3)6n8 + 10n7 16n6 6n5

8x4y3(2x2y2 + 7x5y)16x6y5 + 56x9y4Multiply:

-9m(2m2 7m + 1)

4x2y(3x2 4xy4+ 2y5)

Multiply:

-9m(2m2 7m + 1)-18m3 + 63m2 9m

4x2y(3x2 4xy4+ 2y5)12x4y 16x3y5 + 8x2y6You can extend the distributive property to multiply two binomials, like

(x + 2)(x + 3)

or(3n2 + 5)(2n2 9)

To multiply

essentially you distribute the x and then distribute the 2

To multiply

essentially you distribute the x and then distribute the 2

x2 + 3x

To multiply

essentially you distribute the x and then distribute the 2

x2 + 3x + 2x + 6

x2 + 3x + 2x + 6

To finish it off, you combine the like terms in the middle.

x2 + 3x + 2x + 6

To finish it off, you combine the like terms in the middle.5xThe final answer isx2 + 5x + 6(3n2 + 5)(2n2 9)

(3n2 + 5)(2n2 9)

Distribute 3n2 then distribute 56n4 27n2 + 10n2 45

(3n2 + 5)(2n2 9)

Distribute 3n2 then distribute 56n4 27n2 + 10n2 45Combine like terms-17n2

(3n2 + 5)(2n2 9)

Distribute 3n2 then distribute 56n4 27n2 + 10n2 45Combine like terms-17n26n4 17n2 45 There are lots of ways to remember how the distributive property works with binomials.

x2 + 6x + 4x + 10 = x2 + 10x + 24

The most common mnemonic is called

FOIL

In Gaelic, FOIL is CAID.

However you remember it, its just the distributive property.

Multiply

(3x 5)(2x + 3)

(x3 + 7)(x3 4)Multiply

(3x 5)(2x + 3)6x2 + 9x 10x 15 = 6x2 x 15(x3 + 7)(x3 4)x6 4x3 + 7x3 28 = x6 + 3x3 28Multiply

(2n 5)(3n 6)

(x + 8)(x 8)

Multiply

(2n 5)(3n 6)6n2 12n 15n + 30 = 6n2 27n + 30(x + 8)(x 8)x2 8x + 8x 64 = x2 64Now consider(2x5 + 3)2 and (n 6)2

Now consider(2x5 + 3)2 and (n 6)2

This just means(2x5 + 3)(2x5 + 3)and (n 6)(n 6)

(2x5 + 3)2

(2x5 + 3)(2x5 + 3)

4x10 + 6x5 + 6x5 + 9

4x10 + 12x5 + 9(n 6)2

(n 6)(n 6)

n2 6n 6n + 36

n2 12n + 36Multiply

(x + 4)2

(p3 9)2

Multiply

(x + 4)2= x2 + 8x + 16

(p3 9)2 = p6 18p3 + 81You can extend the distributive property even further

Multiply(3g 3)(2g2 + 4g 4)Multiply(3g 3)(2g2 + 4g 4)

Multiply(x2 + 5)(x2 11x + 6)

Multiply(x2 + 5)(x2 11x + 6)

CHALLENGE:

Multiply (2x2 + x 3)(x2 2x + 5)

CHALLENGE:

Multiply (2x2 + x 3)(x2 2x + 5)