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Here are a few recommended readings before getting started with this lesson.
LHS⋅C=RHS⋅C
Distribute C
Polynomial | Degree | Leading Coefficient |
---|---|---|
P(x)=anxn+an−1xn−1+⋯+a1x+a0 | n | an |
C⋅P(x)=C⋅anxn+C⋅an−1xn−1+⋯+C⋅a1x+C⋅a0 | n | C⋅an |
x=16
Multiply
Subtract terms
Different methods can be used to multiply two polynomials. The following three methods are based on the Distributive Property.
Diego's parents recently bought a piece of land where they plan to raise pigs. They need to fence off a rectangular pigpen before buying the pigs. A farmer friend told Diego that the dimensions of the pigpen, in yards, vary according to the number of pigs x that are being raised in it.
x=15
Calculate power and product
Multiply
Add terms
Izabella bought her nephew a bag of magic grow toys for his birthday. Among the toys, there was a trailer and its rectangular container. Initially, the container was 5 centimeters long, 2 centimeters wide, and 3 centimeters high.
The bag says that when the container is placed underwater, its dimensions will increase according to the following functions. The maximum possible size of the container is reached after being underwater for 9 hours.Distribute -91t2+2t+2
Distribute 41t3,43t2,5t,&15
\CommutativePropMult
am⋅an=am+n
Multiply fractions
Simplify quotient and product
\CommutativePropAdd
Add and subtract terms
t=6
Calculate power
ca⋅b=ca⋅b
Multiply
Calculate quotient
Add and subtract terms
In all the examples seen throughout this lesson, the product of two or more polynomials has resulted in a new polynomial. This is not a coincidence. In fact, the following property guarantees that multiplying polynomials always produces a polynomial.
Given two polynomials P(x) and Q(x), the product P(x)⋅Q(x) is always a polynomial.
Multiplying two polynomials produces a new polynomial.
In other words, the polynomials are closed under multiplication.
Given two polynomials P(x) and Q(x), compute their product and find the required information.
Multiply (x+2) by (x−1)
am⋅an=am+n
a(-b)=-a⋅b
a⋅1=a
Add terms
Distribute 4
Substitute expressions
am⋅an=am+n
Multiply
\CommutativePropAdd
Add and subtract terms