Overview
Polynomials can be multiplied similarly to operations with monomials. There are two different types of properties used, the Distributive Property, and properties of multiplying exponents. In addition, like terms can be combined to simplify the resulting equation.
Multiplying a Monomial by a Polynomial
In some types of expressions, a monomial is multiplied by a polynomial. For example, suppose the problem is to find the area of a rectangle that measures 2x on one side and 4x + 3 on the other side. The product will be 2x(4x + 3). Using the Distributive Property, 2x(4x) + 2x(3) will be the final expression. The first term, 2x(4x), can be simplified to 8x2. The second term, 2x(3), can be simplified to 6x. The final expression is 8x2 + 6x.
Multiplying a Polynomial by a Polynomial
Multiplying a polynomial by another polynomial requires an additional step. Suppose the problem is to find the area of a rectangle that measures 2x + 1 on one side and 3x + 3 on the other side. The product will be a little more complicated, (2x + 1)(3x + 3). It will also use the Distributive Property, as well as properties of exponents.
Use the Distributive Property Twice
The expression can be expanded to (2x + 1)3x + (2x + 1)3. Since there are two terms in the first polynomial and two terms in the second polynomial, the Distributive Property is actually used twice, once for the first polynomial and once for the second polynomial. Solving the expression results in two sets of terms (2x)(3x) + 3x + (2x)3 + 3(1), or 6x2 + 3x + 6x + 3.
Combining Like Terms
The last step in the multiplication is to combine like terms at the same degree. There is only one quadratic term, 6x2, two linear terms 3x + 6x equaling 9x , and one constant, 3. The final expression is 6x2 + 9x + 3.
Interested in algebra tutoring services? Learn more about how we are assisting thousands of students each academic year.
SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Provo, UT visit: Tutoring in Provo, UT