Polynomial Notation: Types, Remainder Theorem, Linear and Quadratic Equation

 A polynomial is an algebraic expression that consists of variables, constants and exponents. The polynomial terms are mostly separated by either addition or subtraction. They can be characterized on the basis of the number of terms and degree of the polynomial. In this article, we will discuss polynomials, their types and the remainder theorem.

Also Read:- Class 10 Degree of polynomial

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Polynomial

Polynomial is an algebraic expression that is made up of coefficients and variables. Variables are often known as indeterminates. The terms ‘poly’ and ‘nominal’ indicate ‘many’ and ‘terms’' respectively.

A polynomial consists of three terms- Constant, Variables and Exponents. 

• Constants- 1, 2, 3, 4, …...etc.
• Variables- x,y,z, a,b,c, etc.
• Exponents- 6 in x⁶, 2 in x³, 2 in x², etc are the examples of exponents 

An example of a polynomial with one variable is x²+x-6. In this, x is a variable, -6 is a constant and 2 is an exponent.

The video below explains this:

Polynomials Detailed Video Explanation:

Also Read: Complex Numbers and Quadratic Equations

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Notation of Polynomial 

The polynomial function is denoted by P(x), where x represents the variable. For example, 

P(x)= x³ - 3x + 2

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Polynomials in Standard Form

A polynomial in standard form is written in the descending power of the variable.

Standard form- ax²+bx+c

Example: Take the polynomial 8x+2+x² and express it in standard form. 

Sol: To present the above polynomial in standard form, we must first determine its degree. The highest degree in the given polynomial is 2. Then we consider if there's a term with a degree less than 2, i.e. 1, and if there is a term with a degree of zero, which is the constant term. 

In standard form, 8x+2+x² can be expressed as- x²+8x+2.

Also Read: Roots of Polynomials

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Terms of a Polynomial

Unlike and Like Terms:

In polynomials, like terms are ones that have the same variable and power. If a polynomial comprises two variables, all powers of the same variable are known as similar terms. Terms with diverse variables and powers are referred to as unlike terms.

Example:

• 9x and 7x are similar/like terms.
• 8y² and 4x⁶ are dissimilar/unlike terms.

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Types of Polynomial

• Based on number of terms:

Polynomials are classed as monomial, binomial, or trinomial on the basis of the number of terms in the expression. 

1. Monomial- A single-term polynomial is known as a monomial. For example- 3x, 4, 9a⁵, etc.
2. Binomial- A binomial polynomial is a polynomial with two terms. For example- -6x+4, 9a⁴+6x, xy² + xy 
3. Trinomial- A trinomial is a polynomial with three terms. For example- -8y⁴+2y+7, x²+5x-6, etc.

• Based on degree of polynomial:

Polynomials can be divided into four categories based on the degree of the polynomial. The degree of the polynomials is the degree of a leading term or the highest power of the variable. The categories include:

1. Zero/Constant polynomial: Polynomials with 0 degree/power.
2. Linear polynomial: Polynomials with its highest exponent being 1.
3. Quadratic polynomial: Polynomials with its highest exponent as 2.
4. Cubic polynomial: Polynomials with its highest exponent as 3

Also Read: Zeros of Polynomial

Division of a Polynomial by a Polynomial

Example: Divide polynomial 5x² + x + 3 with x

Solution: Dividend polynomial: 5x² + x + 3

Divisor Monomial: x.

We have, (5x² + x + 3) / x = (5x²)/x + x/x + 3/x = 5x + 1 + 3/x

3 is not divisible by x. Hence, the division is stopped and 3 is considered to be the remainder.

Therefore, we have 5x² + x + 3 = {x (5x + 1)} + 3.

Result:

Quotient: 5x + 1

Remainder: 3

Since the remainder of the polynomial is not zero, x is not a factor of 5x² + x + 3.

Also Read: 

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Remainder Theorem

The process of dividing one polynomial by another might take a lengthy time. By giving specific rules, the Remainder and Factor Theorems assist us in avoiding the long division procedure.

“Let p (x) be any polynomial of degree one or greater, and ‘a' be any real number. If you divide p (x) by the linear polynomial (x – a), the result is p (a).”

Example: Determine whether q (t) = 6t³ + 6t² – t – 1 is a multiple of 2t + 2.

Solution: If 2t + 2 divides q (t) with a residual of 0, only then q (t) will be a multiple of 2t+2.

2t + 2 = 0, 

t = -2/2 

t = -1

Value of q (t), when t = -1

q (-1)   = 6(-1)³ + 6(-1)² – (-1) – 1.

= 6*(- 1) + 6*1 + 1 – 1

= – 6 + 6

= 0

Since the reminder of q (t) is equal to 0, we can conclude that 2t + 2 is a factor of  6t³ + 6t² – t – 1

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Solving Linear Polynomial

The solution to linear polynomials is simple.To begin, remove the variable term and set the equation equals to zero. Then solve it as a simple algebra problem.

Example: 4x-2, find the value of x

Solution: let, 4x - 2 = 0

4x = 2

x = 2/4

x = 1/2

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Solving Quadratic Polynomial

Rewrite the statement in descending order of degree to solve a quadratic polynomial. Then, to acquire the solution of the problem, equate the equation and execute polynomial factorization.

Example: -14+ 2x² - 7x + x3 = 0

Solution: First arrange the terms in descending order

 x³ + 2x² - 7x - 14 = 0

Now, take the common terms.

x² (x+2) – 7(x+2) =0

(x²-7)(x+2)=0

So, the solutions will be 

x = -2 

x² = 7

x = \(\sqrt{7}\)

Also Read: 

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Things to Remember Based on Polynomials

• Polynomial is an algebraic expression that is made up of coefficients and variables. 
• Standard form a polynomial- P(x)= x³ +ax + b
• The addition or subtraction sign separates two terms in a polynomial.
• In a polynomial, the multiplication and division operations are not used to add extra terms.
• The degree of polynomials should be a whole number.
• Negative exponent expressions are not polynomials.
• A polynomial can be classified into one of three categories based on its terms. They are monomial, binomial, and trinomial, respectively.
• A polynomial can be classified into four types based on its degree. The four types are- zero polynomial, linear polynomial, quadratic polynomial, and cubic polynomial.
• The standard form of a polynomial is written in decreasing order of the power of a variable.