Polynomial equations

As implied by the names, poly means many and nominal means terms, therefore a polynomial is a number of phrases in many different forms.

Polynomials are often composed of a sum or difference of variables and exponents, respectively.

Each phrase in the polynomial represents one of the terms in the polynomial.

Example of a polynomial of the form 4×2+3×7

Rules for determining whether or not an expression is a polynomial

• An algebraic expression should not contain symbols.
• The square root of a number of variables.
• Powers on the variables that are fractional.
• Powers on the variables that are negative.
• Denominators of any fractions can have variables in the denominators.

Moreover, 6x-2 is not a polynomial because it has negative power on the variables.

Because the variable contained within x is a radical, x is not a polynomial.

Because the variable is in the denominator, 1×2 is not a polynomial in this case.

A glossary of terms in polynomial definition- According to polynomial definition, 4×2+3×7 is a prime number.

As shown in the preceding polynomial,

The number 4×2 represents the leading word.

7 is a constant number.

the polynomial has three terms: 3x, 7x, and 7.

What is a Constant?

It is possible to have a constant in a polynomial because it does not contain any variables.

The fact that it does not contain any variables means that its value will never change, which is why it is referred to as a constant word.

As an example, consider the expression 9x + 4, where 4 is the constant because it does not contain any variables.

For example, what is the degree of the following polynomial: 

5×3+4×3+2x+15×3+4×3+2x+15×3+4×3+2x+15×3+4×3+2x+15×3+4×3+2x+1

Solution: The degree of the polynomial in the given situation is four.

Terms of a polynomial according to the polynomial definition are as follows:

“+” and “-” signs separate different parts of the algebraic expression in a polynomial, which are referred to as terms in that polynomial. Each phrase in the polynomial represents one of the terms in the polynomial.

Consider the following polynomial: 4×2+54×2+54×2+ 5,

The number of terms in the polynomial shown in the table below is two. It is possible to classify a polynomial according to the number of terms contained within the polynomial.

Each of these sorts of polynomials has its own unique characteristics.

• Monomial
• Binomial
• Trinomial

What is a Monomial, and how does it work?

A monomial is an algebraic expression that contains only one term and is used to denote a single term in an expression. A monomial expression is defined as follows: if the single term is a non-zero term, then only the algebraic expression is known as such. Some examples of monomials include the following:

• 6x
• 2
• 2a4

What is a Binomial?

An algebraic expression that consists of two terms is known as a binomial polynomial expression. A binomial is generally represented as a sum or difference of two or more monomials. Here are a few examples 

• – 9x+2,
• 6a4+11x
• xy2+4xy

What is a Trinomial?

An algebraic expression that consists of three terms is known as a trinomial polynomial expression. Here are a few examples 

• 4a4+3x+2
• 7×2+8x+5
• 7×3+2×2+7

Polynomial Operations 

The following are the four most important polynomial operations:

• Polynomial Multiplication and Addition
• Polynomial Subtraction is a method of calculating the sum of two polynomials.
• Polynomial Multiplication is a method of multiplying polynomials.
• Polynomial Division is a mathematical concept.

Polynomial Multiplication and Addition

When adding polynomials, always add the terms that are similar, that is, the terms that have the same variable and power. It is always the case that the addition two polynomials results in a polynomial of the same degree. As an illustration,

Example: Find the sum of two polynomials: 9×3+3x2y+4xy−6y2, 3×2+7x2y−9xy+4xy2−5

Polynomial Subtraction

It is a method of calculating the sum of two polynomials.

It is quite similar to adding polynomials, with the main variation being the type of operation that is performed. As a result, to find the solution, subtract the like terms. It should be noted that the subtraction of polynomials results in a polynomial with the same degree as the original polynomial.

Example: Find the difference of two polynomials: x3+3x2y+4xy−6y2, 3×2+5x2y−2xy+4xy2−5

Polynomial Multiplication

It is a method of multiplying polynomials.

When two or more polynomials are multiplied, the outcome is always a polynomial with a greater degree (unless one of them is a constant polynomial). The following is an illustration of polynomial multiplication:

As an illustration, solve (5x-3y)(2x+5y).

Polynomial Division

It is possible that the division of two polynomials will result in a polynomial. Let’s take a closer look at polynomial division in the section below. To divide polynomials, follow the methods outlined below:

Steps in the Polynomial Division are as follows:

When a polynomial has more than one term, we employ the long division approach to solve the problem at hand. 

• Write the polynomial in descending order from highest to lowest.
• Verify that the highest power is used and divide the terms by that number.
• As a division symbol, use the solution from step 2 as a guide.
• Now remove it from the next word and bring it down.
• Steps 2 through 4 should be repeated until there are no more terms to carry down.

Please keep in mind that the final answer, including the remainder, will be in fraction form (last subtract term).

Conclusion

A polynomial function is an expression consisting of a single independent variable, where the variable can appear in the equation more than once with a different degree of the exponent in polynomial, and where the variable can occur in the equation more than once with a different degree of the exponent. Students will also learn how to solve these polynomial functions in this course as well. The graph of a polynomial function can also be depicted utilising turning points, intercepts, end behaviour, and the Intermediate Value Theorem, among other methods of representation.