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 3x 2 (5x 2 – 2x + 3) = 15x 4 – 6x 3 + 9x 2

When you multiply each term, it’s the basic rules of multiplying monomials.  Multiply the coefficients.  Add the exponents.

Multiply: 2n 5 (3n 3 + 5n 2 – 8n – 3) 8x 4 y 3 (2x 2 y 2 + 7x 5 y)

Multiply: 2n 5 (3n 3 + 5n 2 – 8n – 3) 6n n 7 – 16n 6 – 6n 5 8x 4 y 3 (2x 2 y 2 + 7x 5 y) 16x 6 y x 9 y 4

Multiply: -9m(2m 2 – 7m + 1) 4x 2 y(3x 2 – 4xy 4 + 2y 5 )

Multiply: -9m(2m 2 – 7m + 1) -18m m 2 – 9m 4x 2 y(3x 2 – 4xy 4 + 2y 5 ) 12x 4 y – 16x 3 y 5 + 8x 2 y 6

You can extend the distributive property to multiply two binomials, like (x + 2)(x + 3) or(3n 2 + 5)(2n 2 – 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” x 2 + 3x

To multiply essentially you distribute the “x” and then distribute the “2” x 2 + 3x + 2x + 6

x 2 + 3x + 2x + 6 To finish it off, you combine the like terms in the middle.

x 2 + 3x + 2x + 6 To finish it off, you combine the like terms in the middle. 5x The final answer is x 2 + 5x + 6

(3n 2 + 5)(2n 2 – 9)

(3n 2 + 5)(2n 2 – 9) Distribute 3n 2 – then distribute 5 6n 4 – 27n n 2 – 45

(3n 2 + 5)(2n 2 – 9) Distribute 3n 2 – then distribute 5 6n 4 – 27n n 2 – 45 Combine like terms -17n 2

(3n 2 + 5)(2n 2 – 9) Distribute 3n 2 – then distribute 5 6n 4 – 27n n 2 – 45 Combine like terms -17n 2 6n 4 – 17n 2 – 45

There are lots of ways to remember how the distributive property works with binomials. x 2 + 6x + 4x + 10 = x x + 24

The most common mnemonic is called F O I L

In Gaelic, FOIL is CAID.

However you remember it, it’s just the distributive property.

Multiply (3x – 5)(2x + 3) (x 3 + 7)(x 3 – 4)

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

Multiply (2n – 5)(3n – 6) (x + 8)(x – 8)

Multiply (2n – 5)(3n – 6) 6n 2 – 12n – 15n + 30 = 6n 2 – 27n + 30 (x + 8)(x – 8) x 2 – 8x + 8x – 64 = x 2 – 64

Now consider (2x 5 + 3) 2 and (n – 6) 2

Now consider (2x 5 + 3) 2 and (n – 6) 2 This just means (2x 5 + 3)(2x 5 + 3) and (n – 6)(n – 6)

(2x 5 + 3) 2 (2x 5 + 3)(2x 5 + 3) 4x x 5 + 6x x x 5 + 9

(n – 6) 2 (n – 6)(n – 6) n 2 – 6n – 6n + 36 n 2 – 12n + 36

Multiply (x + 4) 2 (p 3 – 9) 2

Multiply (x + 4) 2 = x 2 + 8x + 16 (p 3 – 9) 2 = p 6 – 18p

You can extend the distributive property even further … Multiply (3g – 3)(2g 2 + 4g – 4)

Multiply (3g – 3)(2g 2 + 4g – 4)

Multiply (x 2 + 5)(x 2 – 11x + 6)

CHALLENGE: Multiply (2x 2 + x – 3)(x 2 – 2x + 5)