Multiplying a polynomial and a monomial is a fundamental algebra skill that allows you to simplify larger expressions without expanding them term by term. Now, the process relies on the distributive property, which means distributing the monomial across every term in the polynomial. Once you understand how coefficients and exponents combine, you can multiply expressions accurately, identify like terms, and produce simplified results.
Understanding Monomials and Polynomials
A monomial is an algebraic expression consisting of one term. It may contain a number, one or more variables, or both. Examples include:
- (7)
- (-3x)
- (4x^2)
- (5xy^3)
- (-\frac{2}{5}x^2y)
A polynomial is an expression made up of two or more terms joined by addition or subtraction. Examples include:
- (3x+5)
- (2x^2-4x+7)
- (x^3-2x^2+5x-1)
Every term in a polynomial is itself a monomial. Here's one way to look at it: the polynomial (3x^2-5x+8) contains the monomials (3x^2), (-5x), and (8).
The Distributive Property
The distributive property connects multiplication and addition. It states that multiplying a number or expression by a sum allows you to distribute that multiplication to each individual term.
For example:
[ 2(x+5)=2x+2(5)=2x+10 ]
The same rule applies when multiplying a polynomial by a monomial. If (m) is a monomial and (p_1,p_2,\ldots,p_n) are the terms of a polynomial, then:
[ m(p_1+p_2+\cdots+p_n)=mp_1+mp_2+\cdots+mp_n ]
In simpler words, multiply the monomial by every term in the polynomial Simple as that..
How to Multiply a Polynomial and a Monomial
Follow these steps:
- Identify the monomial and the polynomial.
- Distribute the monomial to each term of the polynomial.
- Multiply coefficients and add exponents of like bases.
- Combine any like terms.
- Write the final answer in standard form.
Example 1: A Basic Example
Multiply:
[ 3x(2x+7) ]
Distribute (3x) to both terms:
[ 3x(2x)+3x(7) ]
Now multiply the coefficients and powers:
[ 6x^2+21x ]
Therefore:
[ \boxed{3x(2x+7)=6x^2+21x} ]
Example 2: A Three-Term Polynomial
Multiply:
[ 4x^2(3x^2-5x+7) ]
Distribute (4x^2) to each term:
[ 4x^2(3x^2)+4x^2(-5x)+4x^2(7) ]
Multiply the coefficients:
[ 12x^4+(-20x^3)+28x^2 ]
Rewrite (-20x^3) as a standard term:
[ \boxed{12x^4-20x^3+28x^2} ]
Multiplying Coefficients and Variables
When multiplying monomials, use two basic rules.
Multiplying Coefficients
Multiply the numerical coefficients:
[ (4x)(6y)=24xy ]
Negative signs are also multiplied:
[ (-3x)(-5y)=15xy ]
Multiplying Variables with the Same Base
When the same variable appears in both monomials, add its exponents:
[ x^2\cdot x^3=x^{2+3}=x^5 ]
For example:
[ (3x^4)(2x^3)=6x^{4+
