To multiply polynomials:
1) Multiply each term in one polynomial by each term in another polynomial
2) Simplify as needed (collect like terms)
1 TERM X 1 TERM
Multiplying a monomial by another monomial:
1) Multiply the constants together
2) Multiply the variables together
3) Combine the result
EXAMPLE:
(2y)(3y)
1) (2)(3) = 6
2) (y)(y) = ?
NOTE: We use brackets to represent multiplication so that we don’t
confuse it with our variable ‘x’
1) Multiply the constants together2) Multiply the variables together3) Combine the result
MULTIPLYING VARIABLES WITH EXPONENTSWe know y² = yy
We know y³ = yyy
So (y²)(y³) = yyyyy = y⁵
However, instead of breaking it down into y’s, we can just add the exponents:
So (y²)(y³) = y²⁺³ = y⁵
Let’s look back at our example
EXAMPLE:
(2y)(3y)
1) (2)(3) = 6
2) (y)(y) = ?
(y)(y) = y¹⁺¹ = y²
3) (2y)(3y) = 6y²
1) Multiply the constants together2) Multiply the variables together3) Combine the result
EXAMPLE:
(4z²)(1/2z³)
1) Multiply the constants together2) Multiply the variables together3) Combine the result
EXAMPLE:
(-6x⁴y)(2xz³)
1) Multiply the constants together2) Multiply the variables together3) Combine the result
EXAMPLE:
(-3ab⁴c²)(-4a²bc³)
1) Multiply the constants together2) Multiply the variables together3) Combine the result
1 TERM X 2 TERMS
We call this the Distributive Property
Multiply the single term by each of the two terms
1 TERM X 3+ TERMS
This is still the distributive property
We will multiply the single term by each of the other terms
a(b + c + d + …) = ab + ac + ad + …
2 TERMS X 3+ TERMS
We can extend the FOIL method to any polynomials being multiplied together
(a + b)(c + d + e) = ac + ad + ae + bc + bd + be
Multiply ‘a’ with all the terms in the second polynomial
Multiply ‘b’ with all the terms in the second polynomial
ANY TERMS X ANY TERMS
This would be true for any number of terms
(a + b + c)(d + e + f) = ad + ae + af + bd + be + bf + cd + ce + cf