MultiplyingPolynomials

Distribute and FOIL

Polynomials * Polynomials

Multiplying a Polynomial by another Polynomial requires more than one distributing step.

Multiply: (2a + 7b)(3a + 5b)

Distribute 2a(3a + 5b) and distribute 7b(3a + 5b):

6a2 + 10ab 21ab + 35b2

Then add those products, adding like terms:

6a2 + 10ab + 21ab + 35b2 = 6a2 + 31ab + 35b2

Polynomials * Polynomials

An alternative is to stack the polynomials and do long multiplication.

(2a + 7b)(3a + 5b)

6a2 + 10ab21ab + 35b2

(2a + 7b)

x (3a + 5b)

Multiply by 5b, then by 3a:(2a + 7b)

x (3a + 5b)When multiplying by 3a, line up the first term under 3a.

+

Add like terms: 6a2 + 31ab + 35b2

Polynomials * Polynomials

Multiply the following polynomials:

Polynomials * Polynomials (x + 5)

x (2x + -1)

-x + -5

2x2 + 10x+

2x2 + 9x + -5

(3w + -2)

x (2w + -5)

-15w + 106w2 + -4w+

6w2 + -19w + 10

Polynomials * Polynomials

(2a2 + a + -1)

x (2a2 + 1)

2a2 + a + -1

4a4 + 2a3 + -2a2+

4a4 + 2a3 + a + -1

Types of Polynomials

• We have names to classify polynomials based on how many terms they have:

Monomial: a polynomial with one term

Binomial: a polynomial with two terms

Trinomial: a polynomial with three terms

F : Multiply the First term in each binomial. 2x • 4x = 8x2

There is an acronym to help us remember how to multiply two binomials without stacking them.

F.O.I.L.

(2x + -3)(4x + 5)

(2x + -3)(4x + 5) = 8x2 + 10x + -12x + -15 = 8x2 + -2x + -15

O : Multiply the Outer terms in the binomials. 2x • 5 = 10x

I : Multiply the Inner terms in the binomials. -3 • 4x = -12x

L : Multiply the Last term in each binomial. -3 • 5 = -15

Use the FOIL method to multiply these binomials:

F.O.I.L.

1) (3a + 4)(2a + 1)

2) (x + 4)(x - 5)

3) (x + 5)(x - 5)

4) (c - 3)(2c - 5)

5) (2w + 3)(2w - 3)

Use the FOIL method to multiply these binomials:

F.O.I.L.

1) (3a + 4)(2a + 1) = 6a2 + 3a + 8a + 4 = 6a2 + 11a + 4

2) (x + 4)(x - 5) = x2 + -5x + 4x + -20 = x2 + -1x + -20

3) (x + 5)(x - 5) = x2 + -5x + 5x + -25 = x2 + -25

4) (c - 3)(2c - 5) = 2c2 + -5c + -6c + 15 = 2c2 + -11c + 15

5) (2w + 3)(2w - 3) = 4w2 + -6w + 6w + -9 = 4w2 + -9