Polynomials

An algebraic expression containing one or more terms is called a polynomial in which exponent of the variables must be a whole number.

2x,  x2−y + yx ,  2y−5yz +2, are polynomials.

2x-2−y , is not a polynomial.

5√yz , is not a polynomial.

Polynomials in One Variable

Polynomials in one variable have only one variable.

2x,

x2 + x,

y3 −5+ 4

Coefficient

The numerical factor (constant) of the term in a polynomial is called it’s coefficient.

In x3 −3x2 + 2x + 5,
the coefficient of  x3 is 1,
the coefficient of  x2 is 3,

the coefficient of  x is 2,
the coefficient of  x0 is 5.

Constant Polynomials

Polynomial having only a constant term is called a constant polynomial.

3,−7, 9 etc.

Zero Polynomial

The constant polynomial 0 is called the zero polynomial.

Denoting a Polynomial in one variable

If the variable in a polynomial is x, we may denote the polynomial by p(x), or q(x), or r(x), etc.

p(x)x3 −3x2 + 2+ 5

q(x) = 3x2 – x + 9

r(y) = 5y6 – 4y2 – 6

s(n) = n2 + n

A polynomial can have any (finite) number of terms.

Monomials

Polynomials having only one term.
p(x) = x2

q(y) = y3

Binomials

Polynomials having two terms.
p(x) = x2+ 2x

q(y) = y3–6

Trinomials

Polynomials having three terms.

p(x) = x3−3x2 + 2

r(n) = 3n2 – n + 9

Degree of a Polynomial

The highest exponent of a variable in a polynomial (with one variable) is called degree of the polynomial.

Coefficient of the term containing highest power of the variable must be non-zero.

The degree of a polynomial 3x7 – 4x6 + x + 9 is 7,

The degree of a polynomial 5y6 – 4y2 – 6 is 6.

The degree of a non-zero constant polynomial is zero.

The degree of the zero polynomial is not defined.

Degree of a Polynomial with more than one variable

Power of variables are added in each term and then compared. Highest sum will be the degree of polynomial.
e.g., 5x2y + 3x2z2 + y3,

Sum of powers in 1st, 2nd & 3rd terms are (2+1), (2+2) & (0+3) respectively. Since, 4 is highest sum, degree of the polynomial is 4.

Linear Polynomial

A polynomial of degree one is called a linear polynomial.

p(x) = x −2
q(y) = 2y + 5

A linear polynomial in one variable will have at most 2 terms.

General form of linear polynomial in x

ax + b,
Where a ≠ 0 and a and b are constants.

A polynomial of degree two is called a quadratic polynomial.

p(x) = x2 −3x 2 + 2
r(n) = 3n2 – n + 9
A quadratic polynomial in one variable will have at most 3 terms.

General form of quadratic polynomial in x

ax2 + bx + c,
Where a ≠ 0 and a, b and c are constants.

Cubic Polynomial

A polynomial of degree three is called a cubic polynomial.
p(x) = 3x3 – 4x2 + x + 9
q(y) = 5y3 – 2y – 6

A cubic polynomial in one variable will have at most 4 terms.

General form of cubic polynomial in x

ax3 + bx2 + cx + d,
Where, a ≠ 0 and a, b, c and d are constants.

Evaluating Polynomials

To find the value of a polynomial at the given value of the variable, we put the value of the variable in the polynomial.

Let’s find the value of p(x) = 3x3 – 4x2 + x + 9 at x =2

To find the value of p (x) at x = 2 we put 2 in the place of x,

p(2) = 3⋅23 – 4⋅22 + 2 + 9
= 3⋅8 – 4⋅4 + 2 + 9
= 24 – 16 + 11
= 19

Let’s find the value of p(x) = x −2 at x =2
p(2) = 2 −2
= 0

Zeroes (Roots) of a Polynomial

Value of the variables for which the value of a polynomial is zero is called zero of the polynomial.

A real number ‘a’ is a zero of a polynomial p(x) if p(a) = 0.

A zero of the polynomial p(x) is also called the root of the polynomial equation, p(x) =0.

Finding the zero of a polynomial p(x), is the same as solving the equation p(x) = 0

Let’s find the zero of p(x) = 2x −4

p(x) = 0
⇒ 2x −4 = 0
⇒ 2x = 4
x = 4/2
x = 2
∴ 2 is the zero of  the polynomial p(x) and, root of the equation p(x)=0.

Root of a linear polynomial in one variable in general form

ax + b = 0 (general form of a linear polynomial in one variable)
⇒ ax = − b
x = − b/a

Every linear polynomial in one variable has a unique zero.

A non-zero constant polynomial has no zero.

e.g., 5x0 = 5 for any value of x.

Every real number is a zero of the zero polynomial.

e.g., 0x = 0 for any value of x.