Lesson Notes: Multiplying Polynomials by Monomials
I. Introduction
In this lesson, we will explore the topic of multiplying polynomials by monomials. Multiplying a polynomial by a monomial involves distributing the monomial to each term in the polynomial. This process requires understanding the properties of multiplication and the rules of combining like terms.
II. Multiplying a Monomial by a Polynomial
To multiply a monomial by a polynomial, we distribute the monomial to each term in the polynomial. This can be done by multiplying the monomial by each term in the polynomial and then combining like terms, if any.
Let's consider an example:
Example 1:
Multiply the monomial 3x by the polynomial 2x^2 + 4xy + 5y.
To find the product, we multiply 3x by each term in the polynomial:
3x(2x^2 + 4xy + 5y) = 6x^3 + 12x^2y + 15xy
The resulting polynomial is obtained by combining the like terms.
III. Simplifying Expressions
Sometimes, after multiplying a monomial by a polynomial, we may need to simplify the expression further by combining like terms. This involves adding or subtracting the coefficients of the like terms.
Example 2:
Simplify the expression 4x(3x^2 + 2xy - 5y^2) - 2x(2x^2 + xy + 3y^2).
To simplify, we distribute the monomials and then combine like terms:
4x(3x^2 + 2xy - 5y^2) - 2x(2x^2 + xy + 3y^2) = 12x^3 + 8x^2y - 20xy^2 - 4x^3 - 2xy - 6y^2
Combining like terms, we get:
(12x^3 - 4x^3) + (8x^2y) + (-20xy^2 - 2xy) + (-6y^2) = 8x^3 + 8x^2y - 22xy^2 - 6y^2
IV. Review Questions
1. Multiply the monomial 5x by the polynomial 3x^2 + 2xy - y^2.
Solution: 15x^3 + 10x^2y - 5xy^2
2. Simplify the expression 2a(3a^2 - 4ab + 5b^2) - 4a(2a^2 + ab - 3b^2).
Solution: 6a^3 - 8a^2b + 10ab^2 - 8a^3 - 4ab + 12b^2
3. Multiply the monomial -2y by the polynomial 4y^3 - 3y^2 + 2y - 1.
Solution: -8y^4 + 6y^3 - 4y^2 + 2y
4. Simplify the expression 3x(2x^2 + 5xy) + 4x(3xy - 2y^2).
Solution: 6x^3 + 15x^2y + 12xy^2 - 8xy^2
5. Multiply the monomial 7a^2b by the polynomial 2ab^2 - 3b^3 + 4.
Solution: 14a^
