Zeros of a Polynomialare those real, imaginary, or complex values that, when put in the polynomial instead of a variable, make the result become zero.
Zeros of a polynomial tell us about the x-intercepts of the polynomial's graph.
For a polynomial P(x), we say that x = a is the zero of the polynomial if P(a) = 0, and all such zeros of a polynomial are commonly called zeros of a polynomial. For example, consider f(x) = 3x - 12. Now, put x = 4 in the polynomial, i.e., f(4) = 3×4 - 12 = 0. Thus, x = 4 is a zero of polynomial f(x) = 3x - 12.
Example: For f(x) = x3 - 6x2 + 11x - 6, is x = 1 zero?
To check whether x = 1 is zero of f(x) = x3 - 6x2 + 11x - 6 or not, put x = 1 in (x)
f(1) = (1)3 - 6×(1)2 + 11×(1) - 6
⇒ f(1) = 1 - 6 + 11 - 6 = 12 -12 = 0
Thus, x = 1 is a zero of f(x).
Zeros of Polynomial Formula
For a linear polynomial of form ax + b, its zero is given by x = \frac{-b}{a}.
For a quadratic polynomial of form ax2 + bx + c, its zero is given by x = \frac{- b ± \sqrt{D}}{2a} where D is Discriminant given by b2 - 4ac.
Finding Zeros of a Polynomial
We can find the zeros of the polynomial for various types of polynomials using various methods that are:
1. For Linear Polynomial
For Linear Polynomials, finding zero is the easiest of all. as there is only one zero and that can also be calculated by simple rearrangement of the polynomial after the equating polynomial to 0.
For example, find zero for linear polynomial f(x) = 2x - 7.
Solution:
To find zero of f(x), equate f(x) to 0.
⇒ 2x - 7 = 0 ⇒ 2x = 7 ⇒ x = 7/2
2. For Quadratic Polynomial
There are various methods to find roots or zeros of a quadratic polynomial such as splitting the middle term, a quadratic formula which is also known as the Shree Dharacharya formula, and completing the square which is somewhat similar to the quadratic formula, as quadratic formula comes from the completing the square for the general quadratic equation.
Example 1: Find out the zeros for P(x) = x2 + 2x - 15.
Example 2: Find the zeros for P(x) = x2 - 16x + 64.
x2 - 16x + 64 = 0
Comparing with ax2 + bx + c = 0,
we get, a = 1, b = -16, and c = 64.
Thus, x = \frac{-(-16) \pm \sqrt{(-16)^2 - 4(1)(64)}}{2(1)}
\Rightarrow x = \frac{16 \pm \sqrt{ 256- 256}}{2}
\Rightarrow x = \frac{16 \pm 0}{2}
⇒ x = 8, 8
3. For Cubic Polynomial
To find zeros of cubic there are many ways, such as rational root theorem and long division together. One method of finding roots of cubic or any higher degree polynomial is as follows:
Step 1: Use the rational root theorem to find the possible roots. i.e., If a polynomial has a rational root it must be the division of p/q, where p is the integer constant and q is the leading coefficient.
Step 2: After finding one root, divide the polynomial with the factor formed by that root using long division and write the polynomial as a product of quotient and dividend.
Step 3: If the quotient is a quadratic expression solve it by the methods above mentioned for quadratic polynomials. If not a polynomial of a degree 2 then repeat steps 1 and 2 until the quotient becomes a polynomial with degree 2.
Step 4: The result of step 3 is the required factors, and by equating the factor to 0, we can find the zeros of the polynomial.
Example: Find the zeros of the cubic polynomial p(x) = x3 - 2x2 - 5x + 6.
p(x) = x3 - 2x2 - 5x + 6
As p/q = 6/1= 6
By rational root theorem, all possible rational roots of the polynomial are divisors of p/q.
Thus, divisors = ±1, ±2, ±3, ±6
x = 1, in p(x), we get
p(1) = (1)3 - 2(1)2 - 5(1) + 6
⇒ p(1) = 1 - 2 - 5 + 6 = 0
Thus, by factor theorem, x - 1 is the factor of p(x).
Thus, x3 - 2x2 - 5x + 6 = (x-1)(x2 -x - 6)
⇒ x3 - 2x2 - 5x + 6 = (x-1)(x-2)(x+3)
For zeroes, p(x) = 0,
Zeros of p(x) are x = 1, x = -2, and x = 3.
Zeros in Graph of Polynomials
In the graph of any polynomial y = f(x), real zeros are the point for which the graph intersects or touches the x-axis. (as a graph with an imaginary zero never cuts the x-axis).
In other words, if there are 3 real solutions of a cubic polynomial then the graph of that cubic polynomial intersects the x-axis three times, but if there is only one real solution for some cubic polynomial then its graph only cuts the x-axis once.
