Zeros of a Polynomial: Geometrical Meaning

The zeros of a polynomial are the values of the variable (usually $x$) that make the polynomial equal to zero. Geometrically, these zeros represent the points where the graph of the polynomial function intersects the x-axis. Each real zero corresponds to an x-intercept. Complex zeros don’t have a direct representation on the x-axis in the standard Cartesian coordinate system.

Formulae

The zeros of a polynomial $P(x)$ are the solutions to the equation:

$P(x) = 0$

For a quadratic polynomial, $ax^2 + bx + c = 0$, we can find the zeros using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$

Examples

Example-1: Consider the quadratic polynomial $P(x) = x^2 – 4$.

To find the zeros, we solve $x^2 – 4 = 0$. Factoring, we get $(x – 2)(x + 2) = 0$. Therefore, the zeros are $x = 2$ and $x = -2$. The graph of this polynomial intersects the x-axis at the points $(2, 0)$ and $(-2, 0)$.

Example-2: Consider the cubic polynomial $P(x) = x^3 – 3x^2 + 2x$.

To find the zeros, we solve $x^3 – 3x^2 + 2x = 0$. Factoring out an $x$, we get $x(x^2 – 3x + 2) = 0$. Further factoring gives $x(x – 1)(x – 2) = 0$. Therefore, the zeros are $x = 0$, $x = 1$, and $x = 2$. The graph intersects the x-axis at $(0, 0)$, $(1, 0)$, and $(2, 0)$.

Theorem with Proof

Theorem: The number of real zeros of a polynomial function $P(x)$ corresponds to the number of x-intercepts on the graph of $P(x)$.

Proof:

By definition, a zero of a polynomial $P(x)$ is a value $c$ such that $P(c) = 0$. When plotting the graph of the polynomial $P(x)$, at the point $(c, P(c))$, the y-coordinate is $P(c) = 0$. This means the point $(c, 0)$ lies on the x-axis. Conversely, if the graph intersects the x-axis at a point $(c, 0)$, then $P(c) = 0$, meaning that ‘c’ is a zero of the polynomial. Each intersection with the x-axis, corresponding to a real zero and vice-versa.

Common mistakes by students

• Confusing Zeros with Solutions of Other Equations: Students may incorrectly associate the zeros of a polynomial with solutions to other equations that might appear related but use different functions.
• Ignoring Repeated Zeros: A repeated zero (e.g., a quadratic with a double root) may appear as one x-intercept where the graph “touches” the x-axis but doesn’t cross it. Students may incorrectly count this as two distinct x-intercepts.
• Focusing solely on the formula without the understanding of the graph: Students memorize formulas, such as the quadratic formula, without visualizing what zeros actually represent graphically which leads to confusion.
• Misinterpreting the nature of complex zeros: Students often fail to realize complex zeros don’t directly intersect the x-axis on a standard Cartesian plane. They might look for a point of intersection, which doesn’t exist.

Real Life Application

Polynomial functions model various real-world phenomena. Finding the zeros is crucial in many applications.

• Engineering: In structural engineering, finding where a beam’s deflection is zero (the x-intercepts) is critical.
• Physics: In projectile motion, the zeros of a quadratic function (representing the trajectory) show the time the projectile hits the ground.
• Economics: Finding the break-even point where revenue equals cost, represented by the zeros of a profit function, is a practical application.

Fun Fact

The Fundamental Theorem of Algebra states that a polynomial of degree $n$ (the highest power of $x$) has exactly $n$ complex zeros (counting multiplicities). This means a polynomial of degree 3 will have 3 zeros, some might be repeated and may not be real numbers. They can be visualized on the complex plane, not directly on the x-axis of a standard Cartesian graph.

Recommended YouTube Videos for Deeper Understanding

Q.1 The graph of a quadratic polynomial $p(x)$ intersects the x-axis at two distinct points. What can you conclude about the zeros of $p(x)$?

Check Solution

Ans: C

The x-intercepts of the graph represent the real zeros of the polynomial. If the graph intersects the x-axis at two distinct points, it means the polynomial has two different real zeros.

Q.2 If the graph of a polynomial $f(x)$ touches the x-axis at only one point and does not cross it, what does this indicate about the zeros of $f(x)$?

Check Solution

Ans: B

When the graph touches but doesn’t cross the x-axis at a point, that point represents a repeated real root, implying the zero has an even multiplicity, specifically 2 in this case.

Q.3 Consider the graph of a cubic polynomial $g(x)$. It intersects the x-axis at $x = -2$, $x = 1$, and $x = 3$. How many real zeros does $g(x)$ have, and what are their values?

Check Solution

Ans: B

The points where the graph intersects the x-axis represent the real zeros. The question explicitly gives these points.

Q.4 A polynomial function, $h(x)$, has no x-intercepts. Which of the following statements is true about the zeros of $h(x)$?

Check Solution

Ans: A

If there are no x-intercepts, this means the graph doesn’t cross the x-axis; thus, the polynomial has no real roots, implying all zeros are complex.

Q.5 A parabola represented by the equation $y = ax^2 + bx + c$ (where $a \neq 0$) has its vertex on the x-axis. How many real zeros does this quadratic function have?

Check Solution

Ans: B

If the vertex lies on the x-axis, it means the parabola touches the x-axis at only one point. This signifies that there is exactly one real zero (a repeated root).

Next Topic: Relationship between Zeros and Coefficients of Quadratic Polynomials