Polynomials

Polynomial: Polynomial with variable x, is denoted by p(x) is expression of the type

${\text{a}}_{\text{n}}{x}^{\text{n}}\text{ + a}{}_{\text{n-1}}x{}^{\text{n-1}}\text{+ a}{}_{\text{n-2}}x{}^{\text{n-2}}\text{ +}\mathrm{...}{\text{+ a}}_{\text{1}}{\text{x + a}}_{\text{0}}\text{.}$

${\text{a}}_{\text{n}}{\text{, a}}_{\text{n-1}}{\text{, a}}_{\text{n-2}}\text{, }\mathrm{...}{\text{, a}}_{\text{0}}$ are the coeeficients, and x is variable. ${\text{a}}_{\text{0}}$ is called constant term.

The polynomial $\text{p}\left(\text{x}\right)$ has $\text{n + 1}$ terms.

e.g. $x^{\text{5}}{\text{ - 5x}}^4{\text{ + 10x}}^3{\text{+7x}}^2\text{-8x + 4}$

Monomials: A polynomial with one term is called monomial.

e.g ${\text{2x, 3x}}^{\text{3}}{\text{, -5y}}^{\text{6}}$

Binomial: A polynomial with two terms is called binomial.

e.g. ${\text{3x + 3x}}^{\text{2}}$

Trinomial: A polynomial with $\text{3}$ terms is called trinomial.

e.g. ${\text{3x + 3x}}^{\text{2}}\text{+ 6}$

Constant polynomial: Aconstant polynomial isa polynomial which is free of variable,and just contains a constant term

e.g. $\text{p}\left(\text{x}\right)\text{ = 5, p}\left(\text{y}\right)\text{ = -7,}$

Degree of polynomial: It is the highest power of variable in the expression of polynomial p(x).

The Polynomial $\text{p}\left(\text{x}\right){\text{ = x}}^{\text{5}}{\text{ - 3x}}^{\text{2}}\text{ + 2x + 4}$ has degree $\text{5}$.

Cubic polynomial: A polynomial with degree $\text{3}$ is called cubic polynomial.

e.g. ${\text{x}}^{\text{3}}{\text{ - 3x}}^{\text{2}}\text{ + 2x + 4}$

Quadratic Polynomial: A polynomial with degree $\text{2}$ is called Quadratic polynomial.

e.g. ${\text{3x}}^{\text{2}}\text{ + 2x + 4}$

Linearpolynomial : A polynomial with degree $\text{1}$ is called Linear polynomial.

e.g. 2x+4

Degree of constant polynomial is zero.

Value of polynomial:

Value obtained after replacing x by a real number k in the expression of $\text{p}\left(\text{x}\right)$ is called value of polynomial at $\text{x = k}$.

For p(x)= ${\text{x}}^{\text{2}}\text{ - 14x + 33}$

Put x=4

We get p(4) = ${\text{4}}^{\text{2}}\text{ - 14 × 4 + 33}$

$\text{= 16 – 56 + 33}$

$\text{p}\left(\text{4}\right)\text{ = -7}$.

We say $\text{-7}$ is the value of polynomial at $\text{x = 4}$.

Zeroes of polynomial:The values of x, at which the polynomial p(x) is equal to 0 is called as zeroes of polynomial.

For e.g. $\text{p}\left(\text{x}\right){\text{ = x}}^{\text{2}}\text{ - 14x + 33}$

We get, $\text{p}\left(\text{11}\right){\text{ = 11}}^{\text{2}}\text{ - 14 × 11 + 33}$

$\text{= 121 - 154 + 33}$

$\text{p}\left(\text{11}\right)\text{ = 0}$ So $\text{11}$ is called zero of the given polynomial.

Remainder theorem:

When polynomial $\text{p}\left(\text{x}\right)$ is divided by polynomial $\text{Q}\left(\text{x}\right)\text{ = x - a }\left(\text{deg Q}\left(\text{x}\right)\text{ < deg p}\left(\text{x}\right)\right)$, then the remainder will be p(a).

e.g.

$\text{p}\left(\text{x}\right){\text{ = 3x}}^{\text{3}}{\text{ - 4x}}^{\text{2}}\text{ + 4x + 5}$

$\text{q}\left(\text{x}\right)\text{ = x - 2}$

If $\text{p}\left(\text{x}\right)\text{ ÷ q}\left(\text{x}\right)$

Remainder= $\text{p}\left(\text{2}\right){\text{ = 3 × 2}}^{\text{3}}{\text{ - 4 × 2}}^{\text{2}}\text{ + 4 × 2 + 5}$

$\text{= 24 – 16 + 8 + 5}$

$\text{= 21}$

Factor theorem:

For polynomial $\text{p}\left(\text{x}\right)$, if $\text{p}\left(\text{a}\right)\text{ = 0}$, then $\left(\text{x - a}\right)$ is factor of $\text{p}\left(\text{x}\right)$

e.g.For $\text{p}\left(\text{x}\right){\text{ = x}}^{\text{2}}\text{ - 14x + 33}$

$\text{p}\left(\text{3}\right)\text{ = 0}$

so(x-3) is factor of ${\text{x}}^{\text{2}}\text{ - 14x + 33}$.

Algebraic identities:

1) ${\left(\text{a + b}\right)}^{\text{2}}{\text{ = a}}^{\text{2}}{\text{ + 2ab + b}}^{\text{2}}$

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

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

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

5) ${\left(\text{a + b + c}\right)}^{\text{2}}{\text{ = a}}^{\text{2}}{\text{ + b}}^{\text{2}}{\text{ + c}}^{\text{2}}\text{ + 2ab + 2bc + 2ac}$

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

Or = ${\text{a}}^{\text{3}}\text{ + 3ab}\left(\text{a + b}\right){\text{ + b}}^{\text{3}}$

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

Or = ${\text{a}}^{\text{3}}\text{ - 3ab}\left(\text{a - b}\right){\text{ - b}}^{\text{3}}$

7) ${\text{a}}^{\text{3}}{\text{ + b}}^{\text{3}}{\text{ + c}}^{\text{3}}\text{ - 3abc = }\left(\text{a + b + c}\right)\left({\text{a}}^{\text{2}}{\text{ + b}}^{\text{2}}{\text{ + c}}^{\text{2}}\text{ - ab - bc - ac}\right)$

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

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