Glossary

5.3 Multiply Polynomials

3 min read•june 24, 2024

Multiplying polynomials is a key skill in algebra. It involves combining simpler expressions to create more complex ones. From multiplying monomials to tackling binomials and beyond, each step builds on the last.

The process uses rules like FOIL and distribution to expand expressions. Special product forms, like perfect squares and differences of squares, offer shortcuts. Understanding structure helps in both multiplication and .

Multiplying Polynomials

Multiplication of monomials

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  • Multiply coefficients of monomials
  • Add exponents of like variables using : xa⋅xb=xa+bx^a \cdot x^b = x^{a+b} (3x2⋅4x3=12x53x^2 \cdot 4x^3 = 12x^5)
  • Simplify resulting by combining

Polynomial by monomial multiplication

  • Distribute monomial to each term of polynomial
    • Multiply of each polynomial term by monomial coefficient
    • Add exponents of like variables using product rule
  • Simplify resulting polynomial by combining like terms (3x2(2x3+4x−1)=6x5+12x3−3x23x^2(2x^3 + 4x - 1) = 6x^5 + 12x^3 - 3x^2)

Binomial multiplication process

  • Use (First, Outer, Inner, Last) to multiply binomials
    1. Multiply first terms
    2. Multiply outer terms
    3. Multiply inner terms
    4. Multiply last terms
    • (x+3)(x−2)=x2−2x+3x−6=x2+x−6(x + 3)(x - 2) = x^2 - 2x + 3x - 6 = x^2 + x - 6
  • Simplify resulting polynomial by combining like terms

Multiplication of varied polynomials

  • Distribute each term of first polynomial to each term of second polynomial
    • Multiply coefficients and add exponents of like variables for each pair of terms
  • Simplify resulting polynomial by combining like terms
    • (2x2+3x−1)(x+2)=2x3+4x2+3x2+6x−x−2=2x3+7x2+5x−2(2x^2 + 3x - 1)(x + 2) = 2x^3 + 4x^2 + 3x^2 + 6x - x - 2 = 2x^3 + 7x^2 + 5x - 2
  • The result is in , showing all terms explicitly

Special product forms

  • Perfect square binomials: (a±b)2=a2±2ab+b2(a \pm b)^2 = a^2 \pm 2ab + b^2 ((x+3)2=x2+6x+9(x + 3)^2 = x^2 + 6x + 9)
  • : (a2−b2)=(a+b)(a−b)(a^2 - b^2) = (a + b)(a - b) ((x2−9)=(x+3)(x−3)(x^2 - 9) = (x + 3)(x - 3))
  • : a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2)
  • : a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Products of polynomial functions

  • Substitute given value for variable in each
  • Multiply resulting polynomials using appropriate methods (monomial, , or general polynomial multiplication)
  • Simplify final result by performing arithmetic operations
    • If f(x)=2x+1f(x) = 2x + 1 and g(x)=x−3g(x) = x - 3, find f(2)⋅g(2)f(2) \cdot g(2)
      1. f(2)=2(2)+1=5f(2) = 2(2) + 1 = 5 and g(2)=2−3=−1g(2) = 2 - 3 = -1
      2. f(2)⋅g(2)=5⋅(−1)=−5f(2) \cdot g(2) = 5 \cdot (-1) = -5

Understanding polynomial structure

  • The of a polynomial is the highest power of the variable in the polynomial
  • The is the term without a variable (e.g., the -2 in 2x3+7x2+5x−22x^3 + 7x^2 + 5x - 2)
  • Factoring is the reverse process of polynomial multiplication, breaking down a polynomial into its factors

Key Terms to Review (19)

Binomial: A binomial is a polynomial expression that consists of two terms, typically connected by addition or subtraction operations. It is a fundamental concept in algebra that is essential for understanding and manipulating polynomial expressions.
Coefficient: A coefficient is a numerical factor that multiplies a variable in an algebraic expression. It represents the number of times a variable appears in a term or an equation.
Constant Term: The constant term is a numerical value that does not have a variable associated with it in a polynomial expression. It is the term that remains unchanged regardless of the value assigned to the variable(s) in the expression.
Degree: The degree of a polynomial is the highest exponent of the variable(s) in the polynomial. It is a measure of the complexity and power of the polynomial expression, and it plays a crucial role in various polynomial operations and equations.
Difference of Cubes: The difference of cubes is a special product in algebra where the difference between two cubes (the third power of a number) can be factored into a simpler expression. This concept is important in understanding how to multiply and factor polynomials, as well as solve polynomial equations.
Difference of Squares: The difference of squares is a special product in algebra where the result of subtracting one perfect square from another perfect square can be factored. This concept is fundamental to understanding polynomial multiplication, factoring trinomials, factoring special products, and solving polynomial equations.
Distributive Property: The distributive property is a fundamental algebraic rule that states that the product of a number and a sum is equal to the sum of the individual products. It allows for the simplification of expressions involving multiplication and addition or subtraction.
Expanded Form: Expanded form is a way of writing a number or polynomial that shows the value of each digit or term. It involves breaking down a number or polynomial into its individual place values or terms to demonstrate the underlying structure and composition.
Exponent: An exponent is a mathematical notation that represents the number of times a base number is multiplied by itself. It is a concise way to express repeated multiplication of the same number.
Factoring: Factoring is the process of breaking down a polynomial expression into a product of simpler polynomial expressions. This technique is widely used in various areas of mathematics, including solving equations, simplifying rational expressions, and working with quadratic functions.
FOIL Method: The FOIL method is a systematic approach used to multiply binomials, where FOIL stands for First, Outer, Inner, Last. It is a widely applied technique in various algebraic operations, including multiplying polynomials, factoring trinomials, and working with special products.
Like Terms: Like terms are algebraic expressions that have the same variable or combination of variables raised to the same power. They can be combined by adding or subtracting their coefficients, as they represent the same quantity in an expression.
Monomial: A monomial is a single algebraic expression consisting of a numerical coefficient, variables, and non-negative integer exponents. It is the most fundamental building block of polynomial expressions, which are central to the topics of adding, subtracting, multiplying, and dividing polynomials.
Multiply Polynomials: Multiplying polynomials is the process of finding the product of two or more polynomial expressions. This involves applying the distributive property and combining like terms to obtain the final result.
Perfect Square Binomial: A perfect square binomial is a special type of polynomial expression that can be factored into the square of a single term. It consists of two terms, one of which is the square of a variable or number, and the other term is twice the product of the square root of the first term and a second term.
Polynomial: A polynomial is an algebraic expression that consists of variables and coefficients, where the variables are raised to non-negative integer powers. Polynomials are fundamental in algebra and play a crucial role in various mathematical topics covered in this course.
Polynomial Function: A polynomial function is an algebraic function that is the sum of one or more terms, each of which is the product of a constant and one or more variables raised to a non-negative integer power. These functions are widely used in mathematics, science, and engineering to model and analyze various phenomena.
Product Rule: The product rule is a fundamental concept in mathematics that describes how to differentiate the product of two or more functions. It is a crucial tool for analyzing and manipulating expressions involving exponents, polynomials, and logarithmic functions.
Sum of Cubes: The sum of cubes is a mathematical expression that represents the sum of two or more numbers raised to the third power. This concept is particularly relevant in the context of multiplying polynomials, factoring special products, and solving polynomial equations.
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