Glossary

10.3 Multiply Polynomials

2 min read•june 25, 2024

Multiplying polynomials is like combining building blocks of math. You'll learn to use the and to multiply expressions with variables and constants. These skills are crucial for solving more complex problems.

By mastering multiplication, you'll be able to tackle real-world scenarios involving areas, volumes, and rates of change. It's a key stepping stone to understanding higher-level math concepts in algebra and beyond.

Multiplying Polynomials

Algebraic Expressions and Components

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  • Polynomials are algebraic expressions consisting of variables, constants, and exponents
  • Variables are letters representing unknown quantities
  • Constants are fixed numerical values in an expression
  • The of polynomials results in a new polynomial
  • involves combining after multiplication

Distribution for polynomial multiplication

  • Distributive property allows multiplying a by each term in a polynomial and adding the results (a(b+c)=ab+aca(b + c) = ab + ac)
  • Multiply the monomial by each term in the polynomial
  • Add the resulting terms to get the final answer
  • Example: 2x(3x2+4x−5)2x(3x^2 + 4x - 5)
    • 2x⋅3x2=6x32x \cdot 3x^2 = 6x^3
    • 2x⋅4x=8x22x \cdot 4x = 8x^2
    • 2x⋅(−5)=−10x2x \cdot (-5) = -10x
    • Final answer: 6x3+8x2−10x6x^3 + 8x^2 - 10x

FOIL method for binomials

  • FOIL (First, Outer, Inner, Last) is a method for multiplying two binomials
  • Given two binomials (a+b)(c+d)(a + b)(c + d), use FOIL:
    • First: Multiply the first terms (a⋅c=aca \cdot c = ac)
    • Outer: Multiply the outer terms (a⋅d=ada \cdot d = ad)
    • Inner: Multiply the inner terms (b⋅c=bcb \cdot c = bc)
    • Last: Multiply the last terms (b⋅d=bdb \cdot d = bd)
  • Add the resulting terms: (a+b)(c+d)=ac+ad+bc+bd(a + b)(c + d) = ac + ad + bc + bd
  • Example: (2x+3)(x−4)(2x + 3)(x - 4)
    • First: 2x⋅x=2x22x \cdot x = 2x^2
    • Outer: 2x⋅(−4)=−8x2x \cdot (-4) = -8x
    • Inner: 3⋅x=3x3 \cdot x = 3x
    • Last: 3⋅(−4)=−123 \cdot (-4) = -12
    • Final answer: 2x2−5x−122x^2 - 5x - 12

Expansion of trinomial-binomial products

  • Use the distributive property twice to multiply a by a
  • Multiply each term of the trinomial by each term of the binomial
  • Add the resulting terms to get the final answer
  • Example: (2x2+3x−1)(x+2)(2x^2 + 3x - 1)(x + 2)
    • Multiply 2x22x^2 by each term in the binomial:
      • 2x2⋅x=2x32x^2 \cdot x = 2x^3
      • 2x2⋅2=4x22x^2 \cdot 2 = 4x^2
    • Multiply 3x3x by each term in the binomial:
      • 3x⋅x=3x23x \cdot x = 3x^2
      • 3x⋅2=6x3x \cdot 2 = 6x
    • Multiply −1-1 by each term in the binomial:
      • −1⋅x=−x-1 \cdot x = -x
      • −1⋅2=−2-1 \cdot 2 = -2
    • Final answer: 2x3+7x2+5x−22x^3 + 7x^2 + 5x - 2

Key Terms to Review (18)

Algebraic Expression: An algebraic expression is a mathematical phrase that combines variables, numbers, and operations to represent a value or relationship. It is a fundamental concept in algebra that allows for the representation and manipulation of quantities without specific numerical values.
Binomial: A binomial is a polynomial expression with exactly two terms. It consists of two monomials, typically with different variables or exponents, connected by an operation such as addition, subtraction, multiplication, or division.
Coefficient: A coefficient is a numerical factor that multiplies a variable in an algebraic expression. It represents the scale or magnitude of the variable, indicating how much of that variable is present in the expression.
Constant: A constant is a quantity in an algebraic expression or equation that has a fixed, unchanging value. It is a value that does not vary or change throughout the given context or problem.
Degree of Polynomial: The degree of a polynomial is the highest exponent of the variable(s) in the polynomial expression. It determines the complexity and behavior of the polynomial function.
Distributive Property: The distributive property is a fundamental algebraic rule that states the product of a number and a sum is equal to the sum of the individual products. It allows for the simplification of expressions by distributing a factor across multiple terms within a parenthesis or other grouping symbol.
Expand: Expanding a polynomial expression involves multiplying the individual terms within the polynomial to create a new, larger polynomial. This process allows for the manipulation and simplification of complex algebraic expressions.
Exponent: An exponent is a mathematical symbol that indicates the number of times a base number is multiplied by itself. It represents repeated multiplication and is used to express large numbers concisely. Exponents are a fundamental concept in algebra and are crucial for understanding and working with expressions, polynomials, and scientific notation.
Factor: A factor is a number or expression that divides evenly into another number or expression without a remainder. Factors are fundamental building blocks used in various mathematical operations and concepts, including multiplication, division, prime factorization, and the distributive property.
FOIL Method: The FOIL method is a systematic approach to multiplying binomials, which are polynomial expressions with two terms. FOIL stands for First, Outer, Inner, Last, which are the steps used to multiply the terms of two binomials together.
Like Terms: Like terms are algebraic expressions that have the same variable(s) raised to the same power. They can be combined by adding or subtracting their coefficients, allowing for the simplification of algebraic expressions.
Monomial: A monomial is a single algebraic expression consisting of a single term, which can include variables, coefficients, and exponents. Monomials are the building blocks of polynomials, which are expressions made up of two or more monomials.
Polynomial: A polynomial is an algebraic expression that consists of variables and coefficients, combined using the operations of addition, subtraction, multiplication, and non-negative integer exponents. Polynomials are fundamental to understanding and working with algebraic expressions, as they form the building blocks for many mathematical concepts and applications.
Product: The product is the result of multiplying two or more numbers or quantities together. It represents the combined or cumulative effect of the factors involved in the multiplication operation.
Simplification: Simplification is the process of reducing an expression or equation to its most basic and concise form without changing its underlying meaning or value. This concept is crucial in various mathematical operations, including working with fractions, mixed numbers, decimals, and polynomials, as it helps to make complex expressions easier to understand, manipulate, and perform further calculations on.
Standard Form: Standard form is a way of expressing numbers, equations, or other mathematical entities in a specific, organized, and easily recognizable format. It provides a consistent and concise way to represent these elements, making them easier to work with, compare, and manipulate across various mathematical contexts.
Trinomial: A trinomial is a polynomial expression with three terms. It is a type of polynomial that can be represented in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers and $a$ is not equal to 0. Trinomials are fundamental in the study of polynomials and play a crucial role in various algebraic operations and factorization techniques.
Variable: A variable is a symbol, typically a letter, that represents an unknown or changeable quantity in an algebraic expression or equation. It is a fundamental concept in algebra that allows for the generalization of mathematical relationships and the solution of problems involving unknown values.
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