Zeros of polynomials

Picture yourself in a video game where you are launching something out into the open with the intent of hitting a target on a surface. The flight path of the projectile creates a lovely, smooth arc in the air. If you were to determine exactly where that object will land, you would be subconsciously trying to solve a classic math problem: determining the zeros of polynomials.

Polynomials play an essential role in algebra and appear in a variety of applications from physics-based models to business prediction models. However, a polynomial by itself is just a mathematical construct that allows us to determine the different possible values for the different variables defined within the polynomial equation. The real wonder of polynomials occurs when we identify the polynomial zeroes (the specific values for the independent variables that will drop the value of the entire polynomial expression to zero).

If you’re a high School student gearing up for boards, a student heading to take a competitive exam and wanting to build a solid foundation, or a person wanting to brush up on previous algebra knowledge, this guide will assist you in understanding the concept of polynomial zeroes in its entirety. You will learn the geometric interpretations of polynomial zeroes, ways to find polynomial zeroes, and how to work through multiple examples of finding polynomial zeroes in a step-by-step manner.

1.0What is a Polynomial? 

A polynomial is an algebraic expression which has variables (i.e., x's), coefficients (i.e., numbers) and powers (i.e., exponents) that can be added, subtracted or multiplied together. All the exponents of the variables in a polynomial must be either whole numbers or non-negative integers (i.e., whole number's), which are 0, 1, 2, 3...

A standard polynomial in a single variable x is typically written as P(x). For example:

• P(x) = 3x + 5 (Linear polynomial)
• P(x) = x² - 5x + 6 (Quadratic polynomial)
• P(x) = 2x³ -3x² + 5x -7(Cubic polynomial)

The highest exponent of the variable in a polynomial is called its degree. The degree is incredibly important because it dictates exactly how many zeroes a polynomial can have.

2.0What Exactly Is a Zero of a Polynomial?

The zero of a polynomial P(x) is a real or complex number K such that when you substitute x = k into the expression, the entire polynomial evaluates to 0.

Formal Definition: A real number k is said to be a zero of a polynomial P(x) if:

P(k) = 0

In different textbooks and exams, you might hear these referred to as the roots of the polynomial equation P(x) = 0, or the solutions to the polynomial. For all practical purposes, when dealing with single variables, "zeroes," "roots," and "solutions" point to the exact same concept.

3.0The Geometric Meaning of Zeroes of a Polynomial

Algebra tells you the numerical values of zeroes, but geometry lets you visualize them. When you plot a polynomial function y = P(x) on a standard Cartesian coordinate plane, it forms a continuous curve.

The zeroes of the polynomial are the x-coordinates of the points where the graph crosses or touches the x-axis.

Why is this the case? Think about it: anywhere along the x-axis, the value of $y$ (which is our $P(x)$) is exactly 0. Therefore, every single intersection point with the x-axis represents a real zero of that polynomial.

Important Insights from Geometry:

• No Intersections = No Real Zeroes: A graph might never cross the x-axis. For instance, the graph of P(x) = x² + 4 floats entirely above the x-axis. This tells us that the polynomial has no real zeroes (it has imaginary or complex zeroes instead).
• Touching vs. Crossing: If a curve turns around right at the x-axis (just touching it at a single point), it represents a repeated or multi-plicity root. For example, P(x) =(x-2)² touches the x-axis precisely at x = 2.

4.0Relationship Between Degree and Number of Zeroes

According to the Fundamental Theorem of Algebra, a polynomial of degree n will have exactly n complex zeroes (which includes both real and imaginary zeroes, counting multiplicities).

However, if we are focusing strictly on real numbers, a polynomial of degree n can have at most n real zeroes.

• A degree 1 polynomial (Linear) will always have exactly 1 real zero.
• A degree 2 polynomial (Quadratic) can have 0, 1, or 2 real zeroes.
• A degree 3 polynomial (Cubic) can have 1, 2, or 3 real zeroes. Note that an odd-degree polynomial will always have at least one real zero because its graph must cross from negative infinity to positive infinity.

5.0How to Find Zeroes of Polynomials (Step-by-Step Methods)

The method you choose to isolate the zeroes depends heavily on the degree and structure of the polynomial you are working with. Here are the primary techniques you need to master.

1. Finding Zeroes of a Linear Polynomial

Linear polynomials are the easiest to solve. You simply set the polynomial to zero and use basic algebraic isolation.

For a standard linear polynomial P(x) = ax + b:

ax + b = 0

ax = -b

$x=-\frac{b}{a}$

2. Finding Zeroes of a Quadratic Polynomial

Quadratic polynomials (ax² + bx + c) are incredibly common in exams. There are two primary methods to crack them:

A. Splitting the Middle Term (Factorization)

This method works beautifully when the roots are clean integers or simple fractions. You need to find two numbers that multiply to give $ac$ (the product of the first and last coefficient) and add up to give $b$ (the middle coefficient).

B. The Quadratic Formula

When a quadratic polynomial defies easy factorization, the quadratic formula is your ultimate backup plan. For any polynomial ax² + bx + c, the zeroes can be directly calculated using:

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

The term under the square root, b² - 4ac, is known as the discriminant $(\Delta )$. It acts as a diagnostic tool for your roots:

• If $(\Delta )>0$ you get two distinct real zeroes.
• If $(\Delta )=0$, you get one repeated real zero.
• If $(\Delta )<0$, you get two complex (imaginary) zeroes.

3. Finding Zeroes of Higher-Degree Polynomials

For cubic or higher polynomials, you typically use a mix of strategies:

• Taking out Common Factors: Always look to see if an x can be factored out immediately.
• Grouping: Group terms pairs to see if a common binomial factor emerges.
• Rational Root Theorem & Synthetic Division: Guess a potential integer root by looking at the factors of the constant term. Once you find one working root (let's say x = c), you divide the polynomial by (x - c) using polynomial long division or synthetic division to reduce it to a simpler quadratic expression.

6.0Relationship Between Zeroes and Coefficients

Mathematicians noticed long ago that you do not even need to solve a polynomial to know how its zeroes behave. There is a direct mathematical link between the zeroes and the coefficients of the polynomial.

For a Quadratic Polynomial (ax² + bx + c)

If $\alpha$ (alpha) and $\beta$ (beta) are the two zeroes of the quadratic expression, then:

• Sum of Zeroes:$\alpha +\beta =-\frac{b}{a}=-\frac{\text{Coefficient of }x}{\text{Coefficient of }{x}^2}$
• Product of Zeroes:$\alpha \times \beta =\frac{c}{a}=\frac{\text{Constant term}}{\text{Coefficient of }{x}^2}$

For a Cubic Polynomial (ax³ + bx² + cx + d)

If$\alpha ,\beta ,and\gamma$ are the three zeroes of a cubic polynomial, then:

• $\alpha +\beta +\gamma =-\frac{b}{a}$
• $\alpha \beta +\beta \gamma +\gamma \alpha =\frac{c}{a}$
• $\alpha \beta \gamma =-\frac{d}{a}$

7.0Solved Examples

Example 1: Find the zero of the linear polynomial P(x) = 5x - 15.

Solution:

To find the zero, set the entire polynomial expression equal to zero:

5x - 15 = 0

Add 15 to both sides of the equation:

5x = 15

Divide both sides by 5:

$x=\frac{15}{5}=3$

Answer: The zero of the polynomial is x = 3.

Example 2: Find the zeroes of the quadratic polynomial$P(x)=x^2-7x+12$ and verify the relationship between the zeroes and its coefficients.

Solution: Step 1: Find the zeroes using the splitting the middle term method.

We need two numbers that multiply to $12(1\times 12)$ and add up to -7. Those numbers are -3 and -4.

$x^2-3x-4x+12=0$

Group the terms to extract common factors:

x(x - 3) - 4(x - 3) = 0

(x - 4)(x - 3) = 0

Set each factor to zero:

• $x-4=0⟹x=4$
• $x-3=0⟹x=3$

So, the zeroes are $\alpha =4and\beta =3.$

Step 2: Verify the relationship with coefficients.

From the polynomial $x^2-7x+12$, we have a = 1, b = -7, and c = 12.

• Sum of zeroes: $\alpha +\beta =4+3=7.$ Using formula: $-\frac{b}{a}=-\frac{-7}{1}=7$ (Verified)
• Product of zeroes: $\alpha \times \beta =4\times 3=12$ Using formula: $\frac{c}{a}=\frac{12}{1}=12$. (Verified)

Example 3: Find all the zeroes of the cubic polynomial $P(x)=x^3-6x^2+11x-6$.

Solution:

Step 1: Use trial and error to find one integer root.

Let us test small integers like 1, -1, 2.

Try x = 1:

$P(1)=(1{)}^3-6(1{)}^2+11(1)-6=1-6+11-6=12-12=0$

Since P(1) = 0, x = 1 is our first zero. This means (x - 1) is a factor of the polynomial.

Step 2: Divide the polynomial to find the remaining quadratic expression.

Dividing$x^3-6x^2+11x-6$ by (x - 1) yields the quotient:

$x^2-5x+6$

Step 3: Solve the resulting quadratic equation.

$x^2-5x+6$

(x - 2)(x - 3) = 0

This gives us x = 2 and x = 3.

Answer: The three zeroes of this cubic polynomial are 1, 2, and 3.

8.0Common Mistakes Students Make

Learning HCF and LCM becomes easier when you avoid these common errors:

• Confusing the Concepts of HCF and LCM: As a rule of thumb, the HCF is related to factors, which are obtained by dividing, and the LCM relates to multiples, which are the result of multiplication.
• Omitting Prime Factors: When the student omits a prime factor, they have made an error in finding their answer.
• Not Enumerating the Complete Set of Multiples: When the student fails to enumerate all the multiples of a set of integers, they will sometimes miss some common multiples.
• Errors in Calculation: This is often caused by using the method of division or because the numbers involved are large.
• Forgetting to Verify: Always check your answer with the formula HCF × LCM = product of the numbers.