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)