Free Algebra
Miscellaneous Equations
Operations with Fractions
Undefined Rational Expressions
Writing Equations for Lines Using Sequences
Intersections of Lines and Conics
Graphing Linear Equations
Solving Equations with Log Terms and Other Terms
Quadratic Expresions - Complete Squares
Adding and Subtracting Fractions with Like Denominators
Multiplying a Fraction by a Whole Number
Solving Equations with Log Terms and Other Terms
Solving Quadratic Equations by Factoring
Locating the Solutions of the Quadratic Equation
Properties of Exponents
Solving Equations with Log Terms on Each Side
Graphs of Trigonometric Functions
Estimating Products and Quotients of Mixed Numbers
The circle
Adding Polynomials
Adding Fractions with Unlike Denominators
Factoring Polynomials
Linear Equations
Powers of Ten
Straight Lines
Dividing With Fractions
Multiplication Property of Equality
Rationalizing Denominators
Multiplying And Dividing Fractions
Distance Between Points on a Number Line
Solving Proportions Using Cross Multiplication
Using the Quadratic Formula
Scientific Notation
Imaginary Numbers
Values of Symbols for Which Fractions are Undefined
Graphing Equations in Three Variables
Writing Fractions as Decimals
Solving an Equation with Two Radical Terms
Solving Linear Systems of Equations by Elimination
Factoring Trinomials
Positive Rational Exponents
Adding and Subtracting Fractions
Negative Integer Exponents
Rise and Run
Multiplying Square Roots
Multiplying Polynomials
Solving Systems of Linear Inequalities
Multiplication Property of Radicals
A Quadratic within a Quadratic
Graphing a Linear Equation
Calculations with Hundreds and Thousands
Multiplication Property of Square and Cube  Roots
Solving Equations with One Log Term
The Cartesian Coordinate Plane
Equivalent Fractions
Adding and Subtracting Square Roots
Solving Systems of Equations
Exponent Laws
Solving Quadratic Equations
Factoring Trinomials
Solving a System of Three Linear Equations by Elimination
Factoring Expressions
Adding and Subtracting Fractions
The parabola
Computations with Scientific Notation
Quadratic Equations
Finding the Greatest Common Factor
Introduction to Fractions
Simplifying Radical Expressions Containing One Term
Polynomial Equations
Graphing and Intercepts
The Number Line
Adding and Subtracting Rational Expressions with Different Denominators
Scientific Notation vs Standard Notation
Factoring by Grouping
Extraneous Roots
Variables and Expressions
Linera Equations
Integers and Substitutions
Squares and Square Roots
Adding and Subtracting Rational Expressions with Different Denominators
Solving Linear Inequalities
Expansion of a Product of Binomials
Powers and Exponents
Finding The Greatest Common Factor
Quadratic Functions
The Intercepts of a Parabola
Solving Equations Containing Rational Expressions
Subtracting Polynomials
Solving Equations
Adding Fractions with Unlike Denominators
Solving Systems of Equations by Substitution
Solving Equations
Product and Quotient of Functions
Try the Free Math Solver or Scroll down to Tutorials!












Please use this form if you would like
to have this math solver on your website,
free of charge.

Multiplying Polynomials

You probably know how to multiply monomials. For example, -2x3 · 4x2 = -8x5.

To multiply a monomial and a polynomial of two or more terms, we apply the distributive property. For example, 3x(x3 - 5) = 3x4 - 15x.


Example 1

Multiplying a monomial and a polynomial

Find the products.

a) 2ab2 · 3a2b

b) (-1)(5 - x)

c) (x3 - 5x + 2)(-3x)


a) 2ab2 · 3a2b = 6a3b3

b) (-1)(5 - x) = -5 + x = x - 5

c) Each term of x3 - 5x + 2 is multiplied by -3x:

(x3 - 5x + 2)(-3x) = -3x4 + 15x2 - 6x

Note what happened to the binomial in Example 1(b) when we multiplied it by -1. If we multiply any difference by -1, we get the same type of result:

-1(a - b) = -a + b = b - a.

Because multiplying by -1 is the same as taking the opposite, we can write this equation as

-(a - b) = b - a.

This equation says that a - b and b - a are opposites or additive inverses of each other. Note that the opposite of a b is -a - b, not a - b.

To multiply a binomial and a trinomial, we can use the distributive property or set it up like multiplication of whole numbers.


Example 2

Multiplying a binomial and a trinomial

Find the product (x + 2)(x2 + 3x - 5).


We can find this product by applying the distributive property twice. First we multiply the binomial and each term of the trinomial:

(x + 2)(x2 + 3x - 5) = (x + 2)x2 + (x + 2)3x + (x + 2)(-5) Distributive property.
  = x3 + 2x2 + 3x2 + 6x - 5x - 10 Distributive property.
  = x3 + 5x2 + x - 10 Combine like terms.

We could have found this product vertically:

  x2 + 3x - 5  
    x + 2  
  2x2 + 6x - 10      2(x2 + 3x - 5) = 2x2 + 6x - 10
x3 + 2x2 - 5x        x(x2 + 3x - 5) = x3 + 3x2 - 5x
x3 + 5x2 + x - 10      Add.


Helpful Hint

Many students find vertical multiplication easier than applying the distributive property twice horizontally. However, you should learn both methods because horizontal multiplication will help you with factoring by grouping.


All Right Reserved. Copyright 2005-2023