# POLYNOMIAL VOCABULARY

On the web page, "Polynomial vocabulary" we are going to see definitions of terms being used in the topic polynomial.

Define polynomial :

A polynomial is an algebraic expression, in which no variables appear in denominators or under radical signs and all variables that do appear are powers of positive integers.

For example,

2x^-1 √y - 4 is not a polynomial

3x³ y² is a polynomial in the variable x and y.

Define terms :

Terms are expressions or numbers that are added or subtracted.

Example :

Find the number of terms in the algebraic expression

2m² + 5m - 3

Solution :

In the given algebraic expression, we have three terms.

2m²  is the first term

5m is the second term

- 3 is the third term

Define degree of the polynomial :

The degree of the highest degree term that appears with non zero coefficients in a polynomial is called degree of the polynomial.

Example :

Find the degree of the polynomial

4 x³ - x + 12

Solution :

The highest power of the given polynomial is 3. Hence the degree of the polynomial is 3.

Define monomial :

We can classify polynomials based on the number of terms. A polynomials which have only one term are known as monomials.

Example :

Find which of the following are monomials

5x - 3 y, 3a, 27m - n + 6, 5x³

Solution :

"3a" and "5x³" are  monomials, because they have only one term

Define binomial :

We can classify polynomials based on the number of terms. A polynomials which have only two terms are called binomials.

Example :

Find which of the following are binomials

5x - 3 y, 3a - 7 - 5x, 27m - n + 6, 5x³ + 7x

Solution :

"5x - 3 y" and "5x³ + 7x" are binomials, because they have only two terms.

Define trinomial :

We can classify polynomials based on the number of terms. A polynomials which have only three terms are called trinomials.

Example :

Find which of the following are trinomials

5x - 3 y, 3a - 7 - 5x, 27m - n + 6, 5x³ + 7x

Solution :

"3a - 7 - 5x" and "27m - n + 6" are tronomials, because they have only three terms. polynomial vocabulary

Define constant polynomial :

We can classify polynomials based on the degree. A polynomial of degree zero is called  a constant polynomial.

General form :

p(x) = c, where c is a real number.

Example :

Find which of the following are constant polynomial

5x, 3a,  7,  27m - n + 6

Solution :

"7" is the constant polynomial.  polynomial vocabulary

Define linear polynomial :

We can classify polynomials based on the degree. A polynomial of degree one is called  a linear polynomial.

General form :

p(x) = ax + b, where a and b are real numbers and "a" is not equal to zero.

Example :

Find which of the following are linear polynomial

5, 3a,  7 - 5x

Solution :

"7 - 5x" and "3a" are linear polynomials.  polynomial vocabulary

We can classify polynomials based on the degree. A polynomial of degree two is called  a quadratic polynomial.

General form :

p(x) = ax² + bx + c, where a, b and c are real numbers and "a" is not equal to zero.

Example :

Find which of the following are quadratic polynomials

2m² + 5m - 3, 5a - 3

Solution :

"2m² + 5m - 3" is the quadratic polynomials.

Define cubic polynomial :

We can classify polynomials based on the degree. A polynomial of degree three is called  a quadratic polynomial.

General form :

p(x) = ax³ + bx² + cx + d, where a, b, c and d are real numbers and "a" is not equal to zero.

Example :

Find which of the following are cubic polynomials

2m² + 5m - 3, 5x³ + 7x, 30m³ + 3m² - 163m + 5

Solution :

"5x³ + 7x", "30m³ + 3m² - 163m + 5" are cubic polynomials.

Define coefficient :

A numerical or constant quantity places before and multiplying the variable in an algebraic expression is called coefficient of the variable.

Example :

Find the coefficients of m² and m of the algebraic expression 2m² + 5m

Solution :

Coefficient of m² = 2

Coefficient of m = 5

Define zero of the polynomial :

If the value of a polynomial is zero for some value of the variable then that value is known as zero of the polynomial

Example :

Find the zero of the polynomial p(x) = x - 2

Solution :

p(x) = x - 2

p(2) = 2 - 2 = 0

Hence 2 is the zero of the given polynomial.

Define roots of a polynomial :

If x = a satisfies the polynomial equation p(x)=0, then x=a is called root of the polynomial p(x)=0.

Example :

Find the roots of the polynomial equation x - 3 = 0

Solution :

Given x - 3 = 0

x = 3

Hence 3 is the root of the polynomial equation.

Define adding and subtracting polynomials :

Adding and subtracting polynomials is the method of combining the like terms.

Example :

Add ( 7p³ +  4p²- 8p + 1 ) and (3p³- 5p²- 10p + 5)

Solution :

Step 1:

The two given polynomials are already in the arranged form.So we can leave it as it is.

=  ( 7p³ + 4p²- 8p + 1) + (3p³ - 5p² - 10p + 5)

Step 2 :

Now we have to write the like terms together starting from the highest power to lowest power.

=  7p³ + 3p³ + 4p²- 5p²- 8p - 10p + 1 + 5

Step 3 :

= 10p³- - 18p + 6

Define multiplying polynomials :

Distributing each term of the first polynomial to every term of the second polynomial is called multiplication of two polynomials.

Example :

Multiply ( - 6 x ) and (- 4 x³)

Solution : = ( - 6 x ) x (- 4 x³)

=  24 x⁴

Define dividing polynomials :

Let p(x) and g(x) be two polynomials such that degree of p(x) > = degree of g(x) and g(x) is not equal to zero.Then there exists unique polynomials q (x ) and r (x ) such that

p(x) = g(x) x q(x) + r(x)

where r(x) = 0 or degree of r(x) < g(x)

Example :

Find the quotient and remainder when 3x²-  4x + 10 is divided by x - 2

Solution : Quotient = 3 x + 2

Remainder = 14

After having gone through the stuff given above, we hope that the students would have understood "Polynomial vocabulary".

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