Polynomials

Intro

An algebraic expression in which variables involved have only non-negative integral powers is called a polynomial.

E.g.- (a) 2x3–4x2+6x–3 is a polynomial in one variable x.

(b) 8p7+4p2+11p3-9p is a polynomial in one variable p.

(c) 4+7x4/5+9x5 is an expression but not a polynomial

since it contains a term x4/5, where 4/5 is not

a non-negative integer.

Degree of a Polynomial in one variable.

• What is degree of the following binomial?

35 2 x 35 2 xThe answer is 2. 5x2 + 3 is a polynomial in x of degree 2.

In case of a polynomial in one variable, the highest power of the variable is called the degree of polynomial.

Degree of a Polynomial in two variables.

• What is degree of the following polynomial?

49375 332 yxyxyx

In case of polynomials on more than one variable, the sum of powers of the variables in each term is taken up and the highest sum so obtained is called the degree of polynomial.

• The answer is five because if we add 2 and 3 , the answer is five which is the highest power in the whole polynomial.

E.g.- is a polynomial in x

and y of degree 7.

92853 243 yxyxyx

Polynomials in one variable

• A polynomial is a monomial or a sum of monomials.

• Each monomial in a polynomial is a term of the polynomial.

The number factor of a term is called the coefficient.

The coefficient of the first term in a polynomial is the lead coefficient.

• A polynomial with two terms is called a binomial. • A polynomial with three term is called a trinomial.

Polynomials in one variable

The degree of a polynomial in one variable is the largest exponent of that variable.

14 x

A constant has no variable. It is a 0 degree polynomial.2This is a 1st degree polynomial. 1st degree polynomials are linear.

1425 2 xx This is a 2nd degree polynomial. 2nd degree polynomials are quadratic.

183 3 x This is a 3rd degree polynomial. 3rd degree polynomials are cubic.

Examples

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Polynomials Degree Classify by degree

Classify by no. of terms.

5 0 Constant Monomial

2x - 4 1 Linear Binomial

3x2 + x 2 Quadratic Binomial

x3 - 4x2 + 1 3 Cubic Trinomial

Standard Form

Phase 1Phase 1 Phase 2Phase 2

To rewrite a polynomial in standard form, rearrange the terms of the polynomial starting with the largest degree term and ending with the lowest degree term.

The leading coefficient, the coefficient of the first term in a polynomial written in standard form, should be positive.

How to convert a polynomial into standard form?

Remainder Theorem

TEXT TEXT TEXT TEXT

Let f(x) be a polynomial of degree n > 1 and let a be any real number.

When f(x) is divided by (x-a) , then the remainder is f(a).

PROOF Suppose when f(x) is divided by (x-a), the quotient is g(x) and the remainder

is r(x).

Then, degree r(x) < degree (x-a)

degree r(x) < 1 [ therefore, degree (x-a)=1]

degree r(x) = 0

r(x) is constant, equal to r (say)

Thus, when f(x) is divided by (x-a), then the quotient is g9x) and the remainder is r.

Therefore, f(x) = (x-a)*g(x) + r (i)

Putting x=a in (i), we get r = f(a)

Thus, when f(x) is divided by (x-a), then the remainder is f(a).

Questions on Remainder Theorem

Q.) Find the remainder when the polynomial

f(x) = x4 + 2x3 – 3x2 + x – 1 is divided by (x-2).

A.) x-2 = 0 x=2

By remainder theorem, we know that when f(x) is divided by (x-2), the remainder is x(2).

Now, f(2) = (24 + 2*23 – 3*22 + 2-1)

= (16 + 16 – 12 + 2 – 1) = 21.

Hence, the required remainder is 21.

Factor Theorem

Let f(x) be a polynomial of degree n > 1 and let a be

any real number.

(i) If f(a) = 0 then (x-a) is a factor of f(x).

PROOF let f(a) = 0

On dividing f(x) by 9x-a), let g(x) be the quotient. Also, by remainder theorem, when f(x) is divided by (x-a), then the remainder is f(a).

therefore f(x) = (x-a)*g(x) + f(a)

f(x) = (x-a)*g(x) [therefore f(a)=0(given]

(x-a) is a factor of f(x).

Algebraic Identities

Some common identities used to factorize polynomials

(x+a)(x+b)=x2+(a+b)x+ab(a+b)2=a2+b2+2ab (a-b)2=a2+b2-2ab a2-b2=(a+b)(a-b)

Algebraic Identities

Advanced identities used to factorize polynomials

(x+y+z)2=x2+y2+z2

+2xy+2yz+2zx

(x-y)3=x3-y3-3xy(x-y)

(x+y)3=x3+y3+3xy(x+y)

x3+y3=(x+y) * (x2+y2-xy) x3-y3=(x+y) *

(x2+y2+xy)

Example

Q.1) Show that (x-3) is a factor of polynomial

f(x)=x3+x2-17x+15.

A.1) By factor theorem, (x-3) will be a factor of f(x) if f(3)=0.

Now, f(x)=x3+x2-17x+15

f(3)=(33+32-17*3+15)=(27+9-51+15)=0

(x-3) is a factor of f(x).

Hence, (x-3) is a factor of the given polynomial f(x).

FAQ

Q.1) Factorize:

(i) 9x2 – 16y2 (ii)x3-x

A.1)(i) (9x2 – 16y2) = (3x)2 – (4y)2

= (3x + 4y)(3x – 4y)

therefore, (9x2-16y2) = (3x + 4y)(3x – 4y)

(ii) (x3-x) = x(x2-1)

= x(x+1)(x-1)

therefore, (x3-x) = x(x + 1)(x-1)

Notes

• A real number ‘a’ is a zero of a polynomial p(x) if p(a)=0. In this case, a is also called a root of the equation p(x)=0.

• Every linear polynomial in one variable has a unique zero, a non-zero constant polynomial has no zero, and every real number is a zero of the zero polynomial.