Multiply each term of one by every term of the other.
Step-by-step explanation:
The distributive property can help you keep it all straight.
(a +b +c)(d +e) = a(d +e) +b(d +e) +c(d +e)
= ad +ae +bd +be +cd +ce
The above shows how each term of one sum is multiplied by every term of the other.
When doing this with polynomials, the rules of arithmetic and exponents apply, so the power of the variable in each product will be the sum of the powers of that variable in each of the terms being multiplied.
Example
(x^2 +x)(x^3 +5) = x^2(x^3 +5) +x(x^3 +5)
= x^5 +5x^2 +x^4 +5x
We have chosen an example that has every product with a different power of x. If there were like terms, they would be combined in the usual way.
With some practice, there are methods available for multiplying longer polynomials without the tedium of writing down lots of intermediate terms. The basic idea is to use mental arithmetic to sum the products that will make up a particular term of the result.
Example
(x^2 +2x +3)(5x^2 +7x +11)
You know this product will have terms ranging from x^4 down to a constant. The x^4 term is the product of the leading terms: (x^2)(5x^2) = 5x^4.
The x^3 term is the sum of products of x^2 and x terms: (x^2)(7x) +(2x)(5x^2) = 17x^3.
The x^2 term is the sum of products of x^2 and constant terms, along with the product of x terms: (x^2)(11) +(2x)(7x) +(3)(5x^2) = 40x^2.
The x-term and constant-term are found in similar fashion. Enough has been shown to give you the flavor of the method.
Perhaps you can see that this understanding will let you write the product of the polynomials very quickly.
Multiplying polynomials is somewhat like multiplying decimal numbers. In multiplying numbers, you separate the product into different powers of 10, but in multiplying polynomials, you keep the products together that apply to the different powers of x.
(12)(35) = (10)(30) +(10)(5) +(2)(30) +(2)(5) = 300 +50 +60 +10 = 420
(x+2)(3x+5) = (x)(3x) +(x)(5) +(2)(3x) +(2)(5) = 3x^2 +11x +10
If x=10, then 3x^2 +11x +10 = 300 +110 +10 = 420, matching the numerical version.
