Polynomial Expressions

Polynomial Expressions
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In this mini lesson we will learn about polynomial expressions, degree of a polynomial, polynomial standard form, zero polynomial, polynomial expressions examples, and parts of a polynomial with solved examples and interactive questions.

The word polynomial is made of two words, "poly" which means 'many' and "nomial", which means terms. It was first used in the seventeenth century and is used in math for representing expressions. 

Henry's teacher asked him whether the given expression was a polynomial expression or not?

Polynomial Expressions: Teacher teaching

Do you want to know the answer?

Stay tuned with Henry to learn more about polynomial expressions!!

Lesson Plan


What Are Polynomial Expressions?

Let us first read about expressions and polynomials

A polynomial is made up of terms, and each term has a coefficient while an expression is a sentence with a minimum of two numbers and at least one math operation in it.

The expressions which satisfy the criterion of a polynomial are polynomial expressions.

Let's see polynomial expressions examples in the following table.

Examples

Examples Polynomial Expressions or not
\(x^2 + 3\sqrt{x} + 1\) No
\(x^2 + \sqrt{3}x + 1\) Yes

In the two cases discussed above, the expression \(x^2 + 3\sqrt{x} + 1\) is not a polynomial expression because the variable has a fractional exponent, i.e., \(\frac{1}{2}\) which is a non-integer value; while for the second expression \(x^2 + \sqrt{3}x + 1\), the fractional power \(\frac{1}{2}\) is on the constant which is 3 in this case, hence it is a polynomial expression.

Standard Form of Polynomial Expressions

The standard form of any polynomial expression is given when the terms of expression are ordered from the highest degree to the lowest degree.

The polynomial standard form can be written as:

\(a_{n}x^{n}+a_{n-1}x^{n-1}+.......+a_{2}x^2+a_{1}x+a_{0}\)

For example, \(ax^2 + bx + c\).

 
tips and tricks
Tips and Tricks

 

  • Positive powers associated with a variable are mandatory in any polynomial, thereby making them one among the important parts of a polynomial.
  • Any expression having a non-integer exponent of the variable is not a polynomial. For example, \(\sqrt{x}\) which has a fractional exponent.
  • A polynomial expression should not have any square roots of variables, any fractional or negative powers on the variables, and no variables should be there in the denominators of any fractions.

What Are the Types of Polynomials?

There are three types of polynomials based on the number of terms that they have:

  • Monomial
  • Binomial
  • Trinomial

Monomial

A monomial consists of only one term with a condition that this term should be non-zero.

Examples: \(6x\), \(7x^3\), \(2ab\)

Binomial 

A binomial is a polynomial that consists of two terms. It is written as the sum or difference of two or more monomials.

Examples: \(2x^4 + 8x\), \(8y^3 + 3x\), \(xy^2 + 3y\)

Trinomial 

A trinomial is a polynomial that consists of three terms.

Examples: \(3x^2 + 4x + 10\), \(5y^4 + 3x^4 + 2x^2y^2\), \(7y^2 + 3y + 17\)


How to Find the Degree of a Polynomial Expression?

For a Single Variable Polynomial

We find the degree of a polynomial expression using the following steps:

  • Step 1: Combine the like terms of the polynomial expression.
  • Step 2: Write the polynomial expression in the standard form.
  • Step 3: Check and select the highest exponent.

The highest exponent of the expression gives the degree of a polynomial.

Let's consider the polynomial expression, 

\(5x^3 + 4x^2 - x^4 - 2x^3 - 5x^2 + x^4\)

In this case, the expression can be simplified as, 

\(3x^3 - x^2\)

Here, the highest exponent corresponding to the polynomial expression is 3

Hence, degree of polynomial expression is 3

For a Multivariable Polynomial

If we take a polynomial expression with two variables, say x and y.

\(x^3 + 3x^2y^4 + 4y^2 + 6\)

We follow the above steps, with an additional step of adding the powers of different variables in the given terms.

Here, the degree of the polynomial is 6

This is because in \(3x^2y^4\), the exponent values of x and y are 2 and 4 respectively.

When we add these, we get 6

Hence, the degree of the multivariable polynomial expression is 6

 
important notes to remember
Important Notes
  • A polynomial with degree 1 is known as a linear polynomial. For example, \(2x + 3\).
  • A polynomial whose degree is 2 is known as a quadratic polynomial. For example, \(x^2 + 4x + 4\).
  • A polynomial with degree 3 is known as a cubic polynomial. For example, \(x^3 + 3x^2 + 3x + 1\).

How to Simplify Polynomial Expressions?

We can simplify polynomial expressions in the following ways:

By combining like terms

The terms having the same variables are combined using arithmetic operations so that the calculation gets easier.

For example, to simplify the polynomial expression,

\(5x^5 + 7x^3 + 8x + 9x^3 - 4x^4 - 10x - 3x^5\)

We combine the like terms to get, 

\(5x^5 - 3x^5 - 4x^4 + 7x^3 + 9x^3 + 8x - 10x \)

On simplifying we get, 

\(2x^5 - 4x^4 + 16x^3 - 2x \)

By using the FOIL technique

The FOIL (First, Outer, Inner, Last) technique is used for the arithmetic operation of multiplication. 

Each step uses the distributive property.

First means multiply the terms which come first in each binomial.

Then, Outer means multiply the outermost terms in the product, followed by the inner terms and then the last terms are multiplied

For example, to simplify the given polynomial expression, we use the FOIL technique,

\((x - 4)(x + 3)\)

The expression can be rewritten as,

\( x (x + 3) - 4 (x + 3)\)

On multiplying the outer terms we get,

\( x^2 + 3x - 4x - 12\)

This expression can be reduced as, 

\( x^2 - x - 12\)


Solved Examples

Example 1

 

 

Help Justin classify whether the expressions given below are polynomials or not.

  • \(\sqrt{x^2+y^2} + 2\)
  • \(x^2 + 3x + 2\)
  • \(\frac{x}{2} + 3x^2 + 5\)
  • \(3x^3 - \sqrt{2}x + 1\)
  • \(\frac{2}{x+3}\)

Solved Examples: Polynomial Expressions

 Solution

Justin will check two things in the given expressions.

  • If the expression has a non-integer exponent of the variable.
  • If the expression has any variable in the denominator.

If an expression has the above mentioned features, it will not be a polynomial expression.

Expressions Criteria to be checked Polynomial or not
\(\sqrt{x^2+y^2} + 2\)  The variables in the expression have a non-integer exponent. No
\(x^2 + 3x + 2\) - Yes
\(\frac{x}{2} + 3x^2 + 5\) - Yes
\(3x^3 - \sqrt{2}x + 1\) - Yes
\(\frac{2}{x+3}\) In this expression, the variable is in the denominator. No
\(\therefore\) Justin used the criteria to classify the expressions.
Example 2

 

 

Which of the following polynomial expressions gives a monomial, binomial or trinomial on simplification? Give the answer in the standard form.

  • \(2x^3 - 10x^3 + 12x^3\) 
  • \(2x^4 - 5x^3 + 9x^3 - 3x^4\)
  • \((x + 3)^2\)

Solved Example: Polynomial Expressions

Solution

Jessica's approach to classify the polynomial expressions after classification would be as follows,

  • \(2x^3 - 10x^3 + 12x^3\)

This expression on simplification gives, \(2x^3 - 10x^3 + 12x^3 = 4x^3\)

The obtained output is a single term which means it is a monomial.

  • \(2x^4 - 5x^3 + 9x^3 - 3x^4\)

This expression on simplification gives, \(2x^4 - 5x^3 + 9x^3 - 3x^4 = 4x^3 - x^4 \)

The obtained output has two terms which means it is a binomial.

In polynomial standard form the obtained expression is written as, \((- x^4 + 4x^3)\)

  • \((x + 3)^2\)

The above expression can be simplified using algebraic identity of \((a+b)^2\)

Hence, the above expression gives the value, \(x^2 - 6x + 9\)

The obtained output has three terms which means it is a trinomial.

The polynomial expression is in its standard form. 

\(\therefore\) All the expressions are classified as monomial, binomial and polynomial.
Example 3

 

 

How will Maria find the product of the polynomial expressions \((2x+6)\) and \((x-8)\)?

Solution

She will write the product of the polynomial expressions as given below,

Solved Examples: Polynomial Expressions

Now to simplify the product of polynomial expressions, she will use the FOIL technique.

Using the distributive property, the above polynomial expressions can be written as,

Solved Examples: Polynomial Expressions

Now applying the FOIL technique, we get,

Solves Examples: Polynomial Expressions

Now combining the like terms we get, 

Solved Examples: Polynomial Expressions

Hence, the product of polynomial expressions \((2x+6)\) and \((x-8)\) on simplification gives, \((2x^2 - 10x - 48)\)

\(\therefore\) Maria simplified the product of polynomial expressions.

Interactive Questions

Here are a few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 

Let's Summarize

We hope you enjoyed understanding polynomial expressions and learning about polynomial, degree of a polynomial, polynomial standard form, zero polynomial, polynomial expressions examples, parts of a polynomial with the practice questions.

The mini-lesson targeted the fascinating concept of polynomial expressions. The math journey around polynomial expressions starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

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At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

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FAQs on Polynomial Expressions

1. What is a polynomial expression?

Any expression which is a polynomial is called a polynomial expression.

2. How do you know if an expression is a polynomial?

If an expression satisfies two criteria:

  • Only the operations of addition, subtraction, multiplication and division by constants is done.
  • The exponents of the variables are non-negative integers.

3. What are the 3 types of polynomials?

The three types of polynomials are:

  • Monomial
  • Binomial 
  • Trinomial

4. What is the difference between a polynomial and an equation?

The difference between a polynomial and an equation is explained as follows:

  • A polynomial is an expression which consists of coefficients, variables, constants, operators and non-negative integers as exponents.
  • An equation is a mathematical statement having an 'equal to' symbol between two algebraic expressions that have equal values.

5. What is a zero polynomial?

A zero polynomial is a polynomial with the degree as 0. It is also called a constant polynomial. 

6. What is an example of a zero polynomial?

The example of a zero polynomial is 3.

7. What are terms in a polynomial?

The terms of polynomials are the parts of the equation which are separated by “+” or “-” signs.

For example, in a polynomial, say, 3x2 + 2x + 4, there are 3 terms.

8. How do you solve polynomial expressions?

The polynomial expressions are solved by:

  • Combining like terms (monomials having same variables using arithmetic operations).
  • Using the FOIL (First, Outer, Inner, Last) technique which is used for arithmetic operation of multiplication.

9. What is the difference between zero polynomial and zero of a polynomial?

A zero polynomial is a polynomial with the degree as 0, whereas, the zero of a polynomial is the value (or values) of variable for which the entire polynomial may result in zero.

10. What is a standard form of a polynomial?

A polynomial is written in its standard form when its term with the highest degree is first, its term of 2nd highest is 2nd, and so on.

It is given as \(a_{n}x^{n}+a_{n-1}x^{n-1}+.......+a_{2}x^2+a_{1}x + a_{0}\).

For example, \(ax^3 + bx^2 + cx + d\).

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