# Zeros Of polynomial

General form of a polynomial in $x$ is $a_n x^n~ +~ a_{n-1} x^{n-1}~ + …..~+ ~a_1 x~+~a_0$, where $a_n,~ a_{n-1},~ …..~,~a_1,~ a_0$ are constants, $a_n~≠~0$ and $n$ is a whole number.

Algebraic expressions such as $√x + x + 5, ~x^2 + \frac{1}{x^2}$ are not polynomials because, all exponents of x in terms of the expressions are not whole numbers.

Degree of a polynomial is highest power of the variable $x$.

• Polynomial of degree 1 is known as linear polynomial.
Standard form is $ax + b$, where $a$ and $b$ are real numbers and $a≠0$.
$2x + 3$ is a linear polynomial.
• Polynomial of degree 2 is known as quadratic polynomial.
Standard form is $ax^2 + bx + c$, where $a,~ b ~and ~c$ are real numbers and $a ≠ 0$
$x^2 + 3x + 4$ is an example for quadratic polynomial.
• Polynomial of degree 3 is known as cubic polynomial.
Standard form is $ax^3 + bx^2 + cx + d$, where $a,~ b,~ c ~and~ d$ are real numbers and $a≠0$.
$x^3 + 4x + 2$ is an example for cubic polynomial.

Similarly,

$y^6 + 3y^4 + y$ is a polynomial in y of degree 6.

Example: What is the value of a if degree of polynomial $x^3 + x^{a-4} + x^2 + 1$ is 4?

Degree of a polynomial $P(x)$ is the highest power of $x$ in $P(x)$.

Therefore, $x^3 + x^{a-4} + x^2 + 1$, $x^{a-4}$ = $x^4$

$a-4$ = $4$, $a$ = $4 + 4$ = $8$

Consider, $P(x)$ = $x^2 – 3x + 2$,

Put $x$ = $3$ in $P(x)$ which gives,

$P(3)$ = $9 – 9 + 2$ = $2$

Replace $x$ by 2 in the polynomial $x^2 – 3x + 2$, which gives $P(2)$ = $4 – 6 + 2$ = $0$.

Similarly, value of $x^2 – 3x + 2$ at $x$ = $0$ is,

$P(0)$ = $0 – 0 + 2$ = $2$

In general; if P(x) is a polynomial in x and k is any real number, then value of $P(k)$ at $x$ = $k$ is denoted by $P(k)$ is found by replacing $x$ by $k$ in $P(x)$.

In the polynomial $x^2 – 3x + 2$,

Replacing $x$ by 1 gives,

$P(1)$ = $1 – 3 + 2$ = $0$

Similarly, replacing $x$ by 2 gives,

$P(2)$ = $4 – 6 + 2$ = $0$

For a polynomial $P(x)$, real number k is said to be zero of polynomial $P(x)$, if
$P(k)$ = $0$.

Therefore, 1 and 2 are the zeros of polynomial $x^2 – 3x + 2$.

Consider, $P(x)$ = $4x + 5$ to be a linear polynomial in one variable;

Let $a$ be zero of $P(x)$, then,

$P(a)$ = $4k + 5$ = $0$

Therefore,$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$ $k$ = $-\frac{5}{4}$

In general, If $k$ is zero of the linear polynomial in one variable; $P(x)$ = $ax + b$, then

$P(k)$ = $ak + b$ = $0$

$k$ = $-\frac{b}{a}$<

It can also be written as,