Glossary

6.5 Polynomial Equations

2 min read•june 24, 2024

Polynomial equations are a powerful tool in algebra, allowing us to solve complex problems. They use the and factoring techniques to find solutions. These equations can represent real-world scenarios, making math applicable to everyday life.

require advanced factoring methods, like the sum and . By mastering these techniques, you'll be able to tackle a wide range of mathematical challenges and practical applications.

Polynomial Equations

Zero Product Property for polynomials

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  • States if the product of two or more factors equals zero, then at least one factor must be zero
  • To find solutions using Zero Product Property:
    1. Factor the polynomial equation
    2. Set each factor equal to zero
    3. Solve the resulting linear equations to find solutions ()
  • Example: In (x−2)(x+3)=0(x - 2)(x + 3) = 0, either x−2=0x - 2 = 0 or x+3=0x + 3 = 0

Factoring of quadratic equations

  • are 2 polynomials in the form ax2+bx+c=0ax^2 + bx + c = 0, where a≠0a \neq 0 ()
  • Factoring methods:
    • : a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b)
    • :
      • a2+2ab+b2=(a+b)2a^2 + 2ab + b^2 = (a+b)^2
      • a2−2ab+b2=(a−b)2a^2 - 2ab + b^2 = (a-b)^2
    • Trial and error
  • After factoring, use Zero Product Property to set factors to zero and solve linear equations

Factoring techniques for higher-degree polynomials

  • Higher-degree polynomials have degree greater than 2
  • Steps to solve by factoring:
    1. Factor out common factors
    2. Look for special factoring patterns:
      • : a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a+b)(a^2 - ab + b^2)
      • Difference of cubes: a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a-b)(a^2 + ab + b^2)
    3. Use or if known factor exists
    4. Apply Zero Product Property to set factors to zero and solve resulting equations

Real-world applications of polynomial equations

  • Identify unknown variable and express with symbol (x)
  • Write equation representing problem using unknown variable
    • Example: Rectangular field length is 5 meters more than twice its width. If area is 135 m^2, find dimensions.
      • Let ww represent field width
      • Length = 2w+52w + 5
      • Area = w(2w+5)=135w(2w + 5) = 135
  • Solve resulting polynomial equation using factoring techniques
  • Interpret solution in problem context, considering constraints
    • In example, width must be positive value

Additional Concepts for Polynomial Equations

  • : An expression consisting of variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer exponents
  • Degree: The highest power of the variable in a polynomial
  • : A value that provides information about the nature of a polynomial's roots
  • : A method for finding potential rational roots of a polynomial equation

Key Terms to Review (16)

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.
Discriminant: The discriminant is a mathematical expression that determines the nature of the solutions to a quadratic equation. It plays a crucial role in understanding the behavior and characteristics of polynomial equations, quadratic equations, and their graphical representations.
Factoring by Grouping: Factoring by grouping is a technique used to factor polynomials by first grouping the terms in the polynomial, then identifying a common factor within each group, and finally factoring out the greatest common factor (GCF) to simplify the expression. This method is particularly useful for polynomials that do not have a clear common factor among all the terms.
Higher-Degree Polynomials: Higher-degree polynomials are algebraic expressions with variables raised to powers greater than one. They are a generalization of linear and quadratic equations, allowing for more complex and versatile functions to be represented and analyzed.
Leading Coefficient: The leading coefficient of a polynomial is the numerical coefficient of the term with the highest degree. It represents the scale or magnitude of the polynomial and plays a crucial role in various polynomial operations and properties.
Perfect Square Trinomials: A perfect square trinomial is a polynomial expression of the form $a^2 + 2ab + b^2$, where $a$ and $b$ are real numbers. These trinomials are called 'perfect' because they can be factored into the square of a binomial expression, $(a + b)^2$.
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.
Polynomial Long Division: Polynomial long division is a method used to divide a polynomial by another polynomial, similar to the long division algorithm used for dividing integers. This technique allows for the division of polynomials and the determination of the quotient and remainder.
Quadratic Equations: A quadratic equation is a polynomial equation of the second degree, where the highest exponent of the variable is 2. These equations take the general form of $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$. Quadratic equations are fundamental in algebra and have numerous applications in various fields, including physics, engineering, and economics.
Rational Root Theorem: The Rational Root Theorem is a powerful tool used in the study of polynomial equations. It provides a method for determining the possible rational roots of a polynomial equation, which can greatly simplify the process of finding the roots of the equation.
Roots: Roots refer to the values of a variable that satisfy an equation or inequality. They represent the solutions to polynomial expressions, where the roots are the x-values that make the equation or inequality equal to zero. Roots are a fundamental concept in algebra, as they are essential for understanding and solving various types of polynomial functions and equations.
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.
Synthetic Division: Synthetic division is a shortcut method for dividing a polynomial by a linear expression of the form (x - a), where 'a' is a constant. It allows for the efficient computation of polynomial division without the need for long division, making it a valuable tool in the context of dividing polynomials and solving polynomial equations.
Zero Product Property: The zero product property states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero. This principle is fundamental in solving various algebraic equations and expressions involving polynomials, rational functions, and radicals.
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