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

A polynomial equation is an equation that has varied terms and generally includes variables coefficient and exponent. Polynomials can retain various exponents. The degree of a polynomial is considered as the greatest exponent. The degree of a polynomial states the number of roots that can be present in a polynomial equation.

A polynomial equation is generally a polynomial expression which has been fixed to make the expression equals to zero.

For Example-If the greatest exponent is 3, then we can say that the polynomial equation has 3 roots. The polynomial equation can be easily written if we are aware of the number of roots.

Polynomials can be solved by factoring them in respect of degrees and variables that exist in the polynomial equation.

4x² + 6x + 21=0 where 4x²+6x+21 is considered as a polynomial expression which is written on the left side and has been fixed to make the polynomial expression equal to zero to form a complete polynomial equation.

Polynomial Equation Formula is generally denoted in the form of anxn where a is considered as a coefficient,x is considered as variable and n is considered as an exponent.

Polynomial Equation Formula in Expanded form can be written in the following manner:

What is known as Polynomial?

Polynomials are algebraic expressions that are composed of two or more algebraic terms. The algebraic terms are Constant, Variables and Exponents. The word poly means "many "and nominal means term in the polynomial.

Let x be the variable as a positive integer and a₀, a₁, a₂ be constants then anxn + an-1xn-1 + an-2xn-2 + a1x +a0 will be considered as polynomial in variable x.

Generally notations such as f(x), g(x), h(x) are used to denote a polynomial in variable x.

F(x) =4x³-2x²+8x-21 and g(x) =7x²-3x+12 are polynomials in variable x whereas p(y) =2y²-3y+4 is a polynomial in a variable y.

7x³-2x²+3√x is not considered as a polynomial because the exponent of x in 3√x is not a positive integer.

Polynomial Function is a type of polynomial equation which retains only one single variable in which a variable can take place in the given polynomial equation much time which varies the degree of the exponent. Polynomial Function graph can be drawn using various elements such as intercepts, end behavior, turning points, and the intermediate value theorem.

F(x) =4x²+6x+21

The exponent of the highest degree term in a polynomial is known as its degree.

For example: - f(x) =4x³-2x²+8x-21 and g(x) =7x²-3x+12 are polynomials of degree 3 and degree 2 respectively.

Based on the degree of a polynomial, there are 5 standard names for Polynomial Equations.

1. Constant Polynomial-A polynomial of degree 0 is called a constant polynomial.

For example-f(x) =2, g(x) = -14, h(y) =5/2 etc are constant polynomials. Constant polynomial 0 or f(x) =o is called the zero polynomial.

2. Linear Polynomial-A polynomial of degree 1 is called a linear polynomial.

For example- f(x)=x-12 ,g(x)=12x,h(x)=-7x+8 are linear polynomials

3. Quadratic Polynomial-A polynomial of degree 2 is known as quadratic polynomial.

For Example-f(x) =2x²-3x+15, g(x) =3/2y²-4y+11/3 etc are quadratic polynomials.

4. Cubic Polynomial-A polynomial of degree 3 is called a cubic polynomial.

For Example-f(x) =12x³-4x²+7x-6,g(x)=7x³+4x-12 are cubic polynomials.

5. BI-Quadratic Polynomial-A polynomial of degree 4 is known as a biquadratic polynomial. For Example-f(x)=12x⁴-7x³+8x²-12c-20 is biquadratic polynomial

Polynomial Equations can be solved with the usage of some general algebraic and factorization rules. The foremost step to solving any of the polynomial equations is to fix 0 on the right side of the polynomial equation.

The foremost step to solving linear polynomials is to make the polynomial equation equals to zero. Then apply some algebraic concepts to solve your equation.

For Example:-

Solve 4x-8

Solution

The first step to make the above equation equals to 0

4X-8=0

4X=8

X=8/4

X=2

Thus the solution of 4x-8 is x=2

The foremost step to solving a quadratic polynomial is to write the polynomial expression in ascending form. Then place the 0 to the right side of the equation.

For Example:-

Solve 4x²-5x+x³-6

The first step is to arrange the above equation in ascending form and place the zero on the right side of the equation.

=x³+4x²-5x-20

Now take the common terms

=X²(x+4)-5(x+4)

=(x²-5)(x+4)

So the required solution will be considered as

X²=5 and x=-4

And x is also equal to √5

1. Find the value of the f (2) and f (-3) in the given Polynomial equation f(x) =2x³-13x²+17x+12.

Solution: - We have f(x) =2x³-13x²+17x+12

f (2)=2*(2)³-13*(2)²*17*2+12

=2*8-13*4+34+12=16-52+34+12=10

f(-3)=2*(3)³-13*(-3)²+17*(-3)+12

=2*27-13*9+17*-3+12

= 54-117-51+12=-210

2. Find the value of polynomial 5x-4x²+3 when x=0.

Solution- Let f(x) = 6x-5x²+2

Putting 0 in place of x, we will get

f(0) =6x (0)-5*(0)²+2

=0-0+2

= 2

Quiz Time

What are the variables in polynomials known as?

a. Symbols

b. Terms

c. Coefficients

d. Degrees

What will be the expanded form of (x+8) (x-10)

a. x²-8x-80

b. x²-2x-80

c. x²+2x+80

d. x²-2x+80

Fun Facts

A linear polynomial may be binomial or monomial. For example- f(x) =7x-15 is a binomial whereas g(x) =3x is a monomial.

A quadratic polynomial may be a binomial or monomial or trinomial. For Example-f(x) =7x² is a monomial, g(x) =2x²+3 is a binomial and h(x) =3x²-2x+4 is a trinomial.

Polynomials can be dealt with more than one variable. For Example-x²+y²+2xyz (where x, y, z are variable. So it is a polynomial with three variables.

FAQ (Frequently Asked Questions)

Define Monomial, Binomial and Trinomial?

Monomial-A monomial is considered as a polynomial with one term. For Example-4x is a polynomial as it has only one term.

Binomial-A binomial is considered as a polynomial with 2 terms. For Example-4x²+5y² is a binomial as it has two terms

Trinomial-A trinomial is considered as a polynomial with 3 terms. For Example-2a²+5a+7 are a trinomial because it has 3 terms.

What are Terms and their Coefficients?

If (f) x) = anxn + an-1xn-1 + an-2xn-2 + a1x +a0 is a polynomial equation with variable x then anxn + an-1xn-1 + an-2xn-2 …..a₁, a₀ and x will be known as terms of polynomial whereas an, an-1 and an-2 will be considered as coefficients in polynomial equations.

The coefficient of an of the highest degree term is known as the leading coefficient whereas a₀ will be known as the constant term.

For Example-f(x) = 2x₂-7x+8

2x₂.-7x and 8 are its terms and 2,-7 and 8 are coefficients of x², x, and constant terms respectively.