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Distribute x^3 + 2x^2 - 3
Multiply
Commutative Property of Addition
Associative Property of Addition
Add terms
Start by drawing a table that has as many rows as there are terms in the first polynomial and that has as many columns as there are terms in the second polynomial.
Polynomial | Number of Terms |
---|---|
P(x) = x^3 + 2x^2 - 3 | 3 |
Q(x) = x^2 + 4 | 2 |
For example, a table with 3 rows and 2 columns is needed to multiply P(x) by Q(x).
Now, write each term of the first polynomial at the left of each cell of the first column. Similarly, write each term of the second polynomial above each cell of the first row.
Start by multiplying the first terms of each binomial. In this case, multiply x by 3x. ( x+6)( 3x-2) = x( 3x) The empty box is there as a reminder that there are still missing terms.
Next, multiply the outer terms — that is, multiply the first term of the left-hand side binomial by the second term of the right-hand side binomial. In this case, multiply x by -2. ( x+6)(3x - 2) = x(3x) + x( -2)
Now multiply the inner terms — that is, multiply the second term of the left-hand side binomial by the first term of the right-hand side binomial. In this case, multiply 6 by 3x. (x+ 6)( 3x-2) = x(3x) + x(-2) + 6( 3x)
Next, multiply the last terms of each binomial — that is, multiply the second term of the left-hand side binomial by the second term of the right-hand side binomial. In this case, multiply 6 by -2. (x+ 6)(3x - 2) = x(3x) + x(-2) + 6(3x) + 6( -2)
Commutative Property of Multiplication
a* a=a^2
a(- b)=- a * b
Multiply
Add terms
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.