Zeros (Roots) of a Polynomial: Concept & Calculation

The zeros (also called roots) of a polynomial are the values of the variable (usually *x*) that make the polynomial equal to zero. In simpler terms, they are the *x*-values where the graph of the polynomial crosses the x-axis (the x-intercepts).

Understanding zeros is crucial for solving polynomial equations, sketching the graph of a polynomial, and analyzing its behavior.

Formulae

The general form of a polynomial is:

$P(x) = a_nx^n + a_{n-1}x^{n-1} + … + a_1x + a_0$

Where:

$P(x)$ represents the polynomial.
$x$ is the variable.
$a_n, a_{n-1}, …, a_1, a_0$ are the coefficients (real numbers).
$n$ is the degree of the polynomial (a non-negative integer).

To find the zeros, we set $P(x) = 0$ and solve for $x$.

Examples

Example-1: Finding the Zeros of a Quadratic Polynomial

Find the zeros of the polynomial $P(x) = x^2 – 5x + 6$.

Solution:

Set $P(x) = 0$: $x^2 – 5x + 6 = 0$
Factor the quadratic equation: $(x – 2)(x – 3) = 0$
Solve for *x*:

$x – 2 = 0 \Rightarrow x = 2$
$x – 3 = 0 \Rightarrow x = 3$

Therefore, the zeros of the polynomial are $x = 2$ and $x = 3$.

Example-2: Finding the Zeros of a Cubic Polynomial

Find the zeros of the polynomial $P(x) = x^3 – 4x^2 + x + 6$.

Solution:

Set $P(x) = 0$: $x^3 – 4x^2 + x + 6 = 0$
By rational root theorem or by observation, we can see that $x=-1$ is a zero. So $(x+1)$ is a factor. We do polynomial division by $(x+1)$ to factorize the polynomial.
Divide $x^3 – 4x^2 + x + 6$ by $(x+1)$ to get the quadratic factor $x^2-5x+6$.
Now the polynomial can be expressed as: $(x+1)(x^2-5x+6) = 0$.
Factor the quadratic equation: $(x – 2)(x – 3) = 0$
Solve for *x*:

$x + 1 = 0 \Rightarrow x = -1$
$x – 2 = 0 \Rightarrow x = 2$
$x – 3 = 0 \Rightarrow x = 3$

Therefore, the zeros of the polynomial are $x = -1$, $x = 2$ and $x = 3$.

Theorem with Proof

The Factor Theorem: A polynomial $P(x)$ has a factor $(x – c)$ if and only if $P(c) = 0$.

Proof:

If part: Assume $(x – c)$ is a factor of $P(x)$. Then we can write $P(x) = (x – c)Q(x)$ for some polynomial $Q(x)$. Substituting $x = c$, we get $P(c) = (c – c)Q(c) = 0 \cdot Q(c) = 0$.

Only if part: Assume $P(c) = 0$. By the Remainder Theorem, when we divide $P(x)$ by $(x – c)$, the remainder is $P(c)$. Since $P(c) = 0$, the remainder is 0. Therefore, $(x – c)$ is a factor of $P(x)$.

Common mistakes by students

Incorrect Factoring: Students often struggle with factoring polynomials, especially higher-degree ones. This leads to incorrect zeros. Practice factoring techniques (e.g., grouping, using the rational root theorem) is crucial.
Forgetting Multiple Zeros: A zero can occur multiple times (e.g., $(x-2)^2$ has a zero of 2 with a multiplicity of 2). Students sometimes miss these repeated roots.
Confusing Zeros and Coefficients: Students might confuse the zeros (solutions for x) with the coefficients of the polynomial.
Incorrect Use of the Quadratic Formula: Students make mistakes when applying the quadratic formula ($x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$) to solve quadratic equations, such as incorrect substitutions or calculation errors.

Real Life Application

Polynomials and their zeros have various real-world applications:

Engineering: Used in designing bridges, buildings, and other structures. Zeros help determine points of stability and stress.
Physics: Used to model projectile motion, wave behavior, and the motion of objects. Zeros can represent points of impact or equilibrium.
Economics: Used to model cost, revenue, and profit functions. Zeros can indicate break-even points or the point where profit is zero.
Computer Graphics: Used in creating curves and surfaces.

Fun Fact

The Fundamental Theorem of Algebra states that a polynomial of degree *n* has exactly *n* complex roots (zeros), counting multiplicity. This means a quadratic equation (degree 2) has two roots, a cubic equation (degree 3) has three roots, and so on. Some of these roots may be real and some may be complex numbers.