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Polynomial Equations

Polynomials are one of the significant concepts of mathematics, and so are Polynomial Equations, where the relation between numbers and variables is explained in a pattern. In math, there are a variety of equations formed with algebraic expressions. Polynomial Equations are also a form of algebraic equations.

Let us learn more about polynomial equations along with their types and the process of solving them.

1. What is a Polynomial Equation?
2. Polynomial Equation Formula
3. Types of Polynomial Equations
4. Solving Polynomial Equations
5. Difference Between Polynomial and Equation
6. FAQs on Polynomial Equation

What is a Polynomial Equation?

A polynomial equation is an equation where a polynomial is set equal to zero. i.e., it is an equation formed with variables, non-negative integer exponents, and coefficients together with operations and an equal sign. It has different exponents. The highest one gives the degree of the equation. For an equation to be a polynomial equation, the variable in it should have only non-negative integer exponents. i.e., the exponents of variables should be only non-negative and they should neither be negative nor be fractions. For example, 2×2 + 3x + 1 is a polynomial and hence 2×2 + 3x + 1 = 0 is a polynomial equation.

Here are more examples of polynomial equations:

In algebra, almost all equations are polynomial equations. Now, let’s explore more details about polynomial equations.

Polynomial Equation Formula

A polynomial equation is always of the form “polynomial = 0”. Algebraically, it is of the form p(x) = an xn + an – 1 xn – 1 + … + a1 x + a0 = 0, where

  • an, an – 1, …., a1, a0 are coefficients and all these numbers are real numbers.
  • ‘x’ is the variable.
  • p(x) means “polynomial in terms of variable x”
  • ‘n’ is a non-negative integer and as it is the highest exponent, it is the degree of p(x).

Polynomial Equation Examples

Here are some examples based on the polynomial equation formula.

Equation Polynomial Equation or Not Reason
x2 + 3√x + 1 = 0 No √x = x1/2 and 1/2 is a non-integer value.
x2 + √3 x + 1 = 0 Yes All powers of x are integers (it is okay if the constants are non-integers)

Think: Determine a few characteristics of an algebraic equation to not to be considered as a polynomial equation.

Types of Polynomial Equations

The type of polynomial equation depends on its degree (the highest exponent of the variable). There are mainly 4 types of polynomial equations:

  • Linear Polynomial Equation
  • Quadratic Polynomial Equation
  • Cubic Polynomial Equation
  • Biquadratic Polynomial Equation

Any polynomial equation other than these is known as a higher degree polynomial equation. Let us see what each of them looks like.

Linear Equations

These are the polynomial equations with degree 1. It is of the form ax + b = 0.

Examples: 2x + 3 = 0, 5x – 7 = 0, etc.

Quadratic Equations

These are the polynomial equations with degree 2. It is of the form ax2 + bx + c = 0.

Examples: 3×2 – 5x + 7 = 0, x2 + 6x + 7 = 0, etc.

Cubic Equations

These are the polynomial equations with degree 3. It is of the form ax3 + bx2 + cx + d = 0.

Examples: x3 – 5 = 0, y3 + 7y2 – 9 = 0, etc.

Biquadratic Equations

These are the polynomial equations with degree 4. It is of the form ax4 + bx3 + cx2 + dx + e = 0.

Example: 3×4 – 5x + 2 = 0.

Solving Polynomial Equations

The process of solving polynomial equation p(x) = 0 is nothing but finding the value(s) of ‘x’ that satisfies the equation. A number ‘a’ is known as a ‘zero’ of a polynomial p(x) if and only if p(a) = 0. Here, ‘a’ is also known as the root of the polynomial equation p(x) = 0. Hence, the process of solving polynomial equations is nothing but finding its roots.

  • To know how to solve linear polynomial equations, click here.
  • To know how to solve quadratic polynomial equations, click here.
  • To know how to solve cubic polynomial equations, click here.

For solving any polynomials other than these, remainder theorem, factor theorem, rational root theorem, and synthetic division are very helpful. Check out each of these topics by clicking on the respective links.

Difference Between Polynomial and Equation

A polynomial is the parent term used to describe a certain type of algebraic expression that contains variables, and constants, and involves the operations of addition, subtraction, multiplication, and division along with only non-negative powers associated with the variables.

Example: 2x + 3

A polynomial equation is a mathematical statement with an ‘equal to’ symbol between two algebraic expressions that have equal values.

Example: 2x + 3 = 7

Important Notes on Polynomial Equations:

  1. The degree of a polynomial equation is the highest power of the variable in the equation.
  2. Solving an equation is finding those values of the variables which satisfy the equation.
  3. You can also find a polynomial equation when roots are known.

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Examples of Polynomial Equations

  1. Example 1: Which of the following are polynomial equations? Justify your answers.

    a) √x + 2 = 0

    b) x2 + 3x + 2 = 0
    c) x/2 + 3×2 + 5 = 0
    d) 3×3 – √2 x + 1 = 0
    e) 2/(x + 3) = 0

    Solution:

    Any equation is NOT a polynomial equation due to one of the following reasons:

    • If the equation has a non-integer (or) negative exponent of the variable.
    • If the equation has any variable in the denominator.

    We will see whether each of the given equations is a polynomial equation or not based on these conditions.

    Equation Polynomial Equation

    (Yes / No)

    Reason a) √x + 2 = 0 No The equation has √x which is equivalent to x1/2, where 1/2 is NOT an integer. b) x2 + 3x + 2 = 0 Yes ___ c) x/2 + 3×2 + 5 = 0 Yes ___ d) 3×3 – √2 x + 1 = 0 Yes ___ e) 2/(x + 3) = 0 No A variable is present in the denominator.

    Answer: Only b), c), and d) are polynomial equations.

  2. Example 2: Which of the following is the polynomial equation 2×4 – 5×3 + 9×2 – 4 = 0? (a) Linear Equation (b) Quadratic Equation (c) Cubic Equation (d) Biquadratic Equation.

    Solution:

    The given polynomial equation is in terms of x. The highest power of x is 4 and hence the degree of the equation is 4. Hence, it is a biquadratic equation.

    Answer: Option (d).

  3. Example 3: Find the polynomial equation of the lowest degree in terms of x whose roots are -3 and 8.

    Solution:

    The roots are -3 and 8. So the corresponding factors are x + 3 and x – 8. Thus, the corresponding polynomial equation is,

    (x + 3) (x – 8) = 0

    x2 – 8x + 3x – 24 = 0

    x2 – 5x – 24 = 0

    Answer: x2 – 5x – 24 = 0.

Practice Questions on Polynomial Equation

FAQs on Polynomial Equation

How Will You Know if an Equation is a Polynomial Equation?

A polynomial equation is basically a polynomial expression equated to 0. For example, 3×2 – 5 = 0 is a polynomial equation as 3×2 – 5 is a polynomial expression.

What is the Difference Between a Polynomial and a Polynomial Equation?

A polynomial is an expression that is made up of one or more variables, coefficients, and non-negative integer exponents of variables. An equation is a mathematical statement with an ‘equal to’ symbol between two algebraic expressions that have equal values. Thus, a polynomial equation is an equation that is of the form polynomial = 0.

What are the Different Types of Polynomial Equations?

The different types of polynomial equations are – linear equations, quadratic equations, cubic equations, and biquadratic equations.

What is Polynomial Equation Formula?

A polynomial formula is a polynomial function set to 0 and is of the form p(x) = an xn + an – 1 xn – 1 + … + a1 x + a0 = 0.

What is Not a Polynomial Equation?

Any algebraic equation with a negative exponent or fractional exponent is NOT ot a polynomial equation. In other words, if an equation that has “= 0” in it doesn’t have a polynomial in it, then it is NOT a polynomial equation.

What is the General Form of a Polynomial Equation?

The general form of polynomial equation in terms of x is an xn + an – 1 xn – 1 + … + a1 x + a0 = 0. Here, an, an – 1, …., a1, a0 are known as coefficients and these are real numbers.

How do You Solve Polynomial Equations?

The polynomial equations can be solved by factoring them and setting each factor to zero. Also, we can graph the left side of the polynomial equation p(x) = 0 using a graphing calculator and in that case, the x-intercepts of the graph would give the roots of the polynomial equation.

How to Find the Degree of Polynomial Equations?

The highest power of the variable term in the polynomial is the degree of the polynomial. For example, the degree of the polynomial equation x3 + 2x + 5 = 0 is 3.

How do You Find the Roots of a Polynomial Equation?

The roots of a polynomial equation can be found using one of the following methods:

  • We first find one root either by trial and error method or by using the rational root theorem. Then we use the corresponding factor and divide the given polynomial to find the other roots.
  • We can find all the roots by completely factorizing the polynomial in the given equation (if possible) and by setting each factor to zero.
  • We can just graph the polynomial and its x-intercepts would be the roots of the equation.

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