### Polynomials - Revision Notes

CBSE Class 09 Mathematics

Revision Notes
CHAPTER – 2
POLYNOMIALS

1. Polynomials in one Variable
2. Zeroes of a Polynomial
3. Remainder Theorem
4. Factorisation of Polynomials
5. Algebraic Identities

Constants : A symbol having a fixed numerical value is called a constant.

Variables : A symbol which may be assigned different numerical values is known as variable.

Algebraic expressions : A combination of constants and variables connected by some or all of the operations +, -, *,/  is known as algebraic expression.

Terms : The several parts of an algebraic expression separated by '+' or '-' operations are called the terms of the expression.

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

(i) $5{x}^{2}-4{x}^{2}-6x-3$ is a polynomial in variable x.

(ii) $5+8{x}^{\frac{3}{2}}+4{x}^{-2}$ is an expression but not a polynomial.

Polynomials are denoted by p(x), q(x) and r(x) etc.

Coefficients : In the polynomial ${x}^{3}+3{x}^{2}+3x+1,$, coefficient of ${x}^{3},\phantom{\rule{thickmathspace}{0ex}}{x}^{2},\phantom{\rule{thickmathspace}{0ex}}x\phantom{\rule{thickmathspace}{0ex}}are\phantom{\rule{thickmathspace}{0ex}}1,\phantom{\rule{thickmathspace}{0ex}}3,\phantom{\rule{thickmathspace}{0ex}}3$ respectively and we also say that +1 is the constant term in it.

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

A polynomial of degree n has n roots.

Classification of polynomials on the basis of degree.

 degree Polynomial Example (a) 1 Linear x+1,2x+3etc.$x+1,\phantom{\rule{thickmathspace}{0ex}}2x+3\phantom{\rule{thinmathspace}{0ex}}etc.$ (b) 2 Quadratic ax2+bx+cetc.$a{x}^{2}+bx+c\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}etc.$ (c) 3 Cubic x3+3x2+1etc.${x}^{3}+3{x}^{2}+1\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}etc.$ (d) 4 Biquadratic x4−1${x}^{4}-1$

Classification of polynomials on the basis of number of terms

 No. of terms Polynomial & Examples. (i) 1 Monomial - 5,3x,13yetc.$5,3x,\frac{1}{3}y\phantom{\rule{thickmathspace}{0ex}}etc.$ (ii) 2 Binomial - (3+6x),(x−5y)$\left(3+6x\right),\phantom{\rule{thickmathspace}{0ex}}\left(x-5y\right)$ etc. (iii) 3 Trinomial- 2x2+4x+2etc.$2{x}^{2}+4x+2\phantom{\rule{thickmathspace}{0ex}}etc.$

Constant polynomial : A polynomial containing one term only, consisting a constant term is called a constant polynomial.The degree of non-zero constant polynomial is zero.

Zero polynomial : A polynomial consisting of one term, namely zero only is called a zero polynomial.

The degree of zero polynomial is not defined.

Zeroes of a polynomial : Let $p\left(x\right)$ be a polynomial. If $p\left(a\right)$=0, then we say  that "a"  is a zero of the polynomial of p(x).

Remark : Finding the zeroes of polynomial p(x) means solving the equation p(x)=0.

Remainder theorem : Let $f\left(x\right)$ be a polynomial of degree $n⩾1$ and let a be any real number. When f(x) is divided by $\left(x-a\right)$ then the remainder is f ( a)

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

If f(a) = 0 then, (x – a) is factor of f(x)

If f(x – a) is factor of f(x) then f(a) = 0

Factor : A polynomial $p\left(x\right)$ is called factor of $q\left(x\right)$ divides $q\left(x\right)$ exactly.

Factorization : To express a given polynomial as the product of polynomials each of

degree less than that of the given polynomial such that no such a factor has a factor of

lower degree, is called factorization.

Some algebraic identities useful in factorization:

(i) $\left(x+y{\right)}^{2}$$\left(x{\right)}^{2}$+2xy+$\left(y{\right)}^{2}$

(ii) $\left(x-y{\right)}^{2}$$\left(x{\right)}^{2}$-2xy+$\left(y{\right)}^{2}$

(iii) ${x}^{2}$-${y}^{2}$=(x-y)(x+y)

(iv) (x+a)(x+b)= $\left(x{\right)}^{2}$+(a+b)x+ab

(v)$\left(x+y+z{\right)}^{2}$=${x}^{2}+$${y}^{2}+$${z}^{2}$+2xy+2yz+2zx

(vi) $\left(x+y{\right)}^{3}$=${x}^{3}$+${y}^{3}$+3xy(x+y)

(vii) $\left(x-y{\right)}^{3}$=${x}^{3}$-${y}^{3}$-3xy(x-y)

(viii) ${x}^{3}$ + ${y}^{3}$ + ${z}^{3}$ - 3xyz =(x+y+z) (${x}^{2}$ + ${y}^{2}$ + ${z}^{2}$ - xy - yz - zx)

${x}^{3}$+${y}^{3}$+${z}^{3}$=3xyz if x+y+z=0

(ix) ${a}^{3}+{b}^{3}=\left(a+b\right)\left({a}^{2}-ab+{b}^{2}\right)$

(x)