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 wouldn’t 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 + 9x2

When you multiply each term, it’s 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 + 56x9y4

Multiply:

-9m(2m2 – 7m + 1)

4x2y(3x2 – 4xy4+ 2y5)

Multiply:

-9m(2m2 – 7m + 1)-18m3 + 63m2 – 9m

4x2y(3x2 – 4xy4+ 2y5)12x4y – 16x3y5 + 8x2y6

You 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 is

x2 + 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 – 45

Combine like terms-17n2

(3n2 + 5)(2n2 – 9)

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

Combine like terms-17n2

6n4 – 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, it’s 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 – 28

Multiply

(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 – 64

Now 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 + 36

Multiply

(x + 4)2

(p3 – 9)2

Multiply

(x + 4)2

= x2 + 8x + 16

(p3 – 9)2

= p6 – 18p3 + 81

You 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)