# Zeros of Polynomial

Zeros of Polynomial : It is a solution to the polynomial equation, P(x) = 0. It is that value of x that makes the polynomial equal to 0.The value of a polynomial f(x) at x = a is obtained by substituting x = a in the given polynomial and is denoted by f(a) which is Zeros of Polynomial.

**Examples Zeros of Polynomial :**

Find the value of f(x) = x

^{3}- 2x

^{2}+ 2x – 4 at x = -1

**Solution:**

f (x) = x

^{3}- 2x

^{2}+ 2x – 4

f(-1) = (-1)

^{3}- 2(-1)

^{2}+ 2(-1) – 4

= -1 -2 – 2 – 4

= - 9

**Note : If f(a) = 0 then x = a is the root of the polynomial f(x).**

**Relationship between the zeroes and coefficient of a Polynomial**

**Case 1 : Quadratic Polynomial**

ax

^{2}+bx + c

If α and β are the roots of the equation then

Sum of zeroes = α + β = Product of zeroes = αβ = |

From the above the equation of polynomial is given by

**x**

^{2}-( α + β) x + αβ**Examples :**

1) Write the equation of the polynomial, if sum of zeroes = -8 and product of zeroes is 15

**Solution**: As Sum of zeroes = α + β = - 8

Product of zeroes = αβ = 15

Equation is x

^{2}-( α + β) x + αβ

x

^{2}- ( - 8) x + 15

x

^{2}+ 8x + 15

**Case 2 : Cubic Polynomial**

ax

^{3}+ bx

^{2}+ cx + d , a ≠0

Sum of zeroes = α + β+γ = - b/a Sum of the product of zeroes taken two at a time = αβ + βγ+αγ = c/a Product of zeroes = αβγ = -d/a |

The cubic polynomial can be written as

**x**

^{3}- (α + β+γ)x^{2}+ (αβ + βγ+αγ)x - αβγ**Example :**

1) Find the cubic polynomial with the sum, sum of the product of zeroes taken two at a time, and product of its zeroes as 2,-7 ,-14 respectively.

**Solution :**

If α,β and γ are the zeroes of a cubic polynomial then

x

^{3}- (α + β+γ)x

^{2}+ (αβ + βγ+αγ)x - αβγ

α + β+γ = 2

αβ + βγ+αγ = -7

αβγ = -14

x

^{3}-(2 )x

^{2}+ (-7)x – (-14)

x

^{3}-2 x

^{2}-7x + 14

**Polynomial**

• Degree of the Polynomial

• Zeros of Polynomial

• Remainder Theorem

• Find remainder by Synthetic Division

• Rational root test in Polynomial

• Solved Examples on Polynomial

• Degree of the Polynomial

• Zeros of Polynomial

• Remainder Theorem

• Find remainder by Synthetic Division

• Rational root test in Polynomial

• Solved Examples on Polynomial

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