HCF is the greatest number that divides two or more numbers exactly, whereas LCM is the smallest number divisible by the given numbers.
For two numbers: HCF x LCM = Product of the Numbers
Write the prime factors of each number and multiply the common prime factors with the smallest powers.
LCM is used in timetable synchronization, fractions, scheduling events, and solving repeated cycle problems.
Yes. When the two numbers are equal, their HCF and LCM are also equal
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Class 10 Maths: 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).
Class: 10 Maths (CBSE)
Chapter: Polynomials
Estimated Learning Time: 15–20 Minutes
1.0Learning Outcomes
After studying this chapter, students will be able to:
Understand the concept of zeroes of a polynomial and identify the relationship between zeroes and factors of polynomials.
Find the zeroes of quadratic polynomials using different methods and verify the relationship between zeroes and coefficients.
Apply algebraic concepts to solve problems involving polynomial equations and their graphical representation.
Develop problem-solving skills by practicing important questions based on polynomial concepts as per the CBSE Class 10 syllabus.
2.0What 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.
What 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.
The 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.
Polynomial Type
Degree
Shape of the Graph
Maximum X-Axis Intercepts (Real Zeroes)
Linear
1
A straight line
Exactly 1
Quadratic
2
A U-shaped curve (Parabola)
Up to 2
Cubic
3
An S-like wave curve
Up to 3
Biquadratic
4
A W-shaped or M-shaped curve
Up to 4
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.
Relationship 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.
How 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=−ab
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=2a−b±b2−4ac
The term under the square root, b2 - 4ac, is known as the discriminant(Δ). It acts as a diagnostic tool for your roots:
If (Δ)>0 you get two distinct real zeroes.
If (Δ)=0, you get one repeated real zero.
If (Δ)<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.
Relationship 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:α+β=−ab=−Coefficient of x2Coefficient of x
Product of Zeroes:α×β=ac=Coefficient of x2Constant term
For a Cubic Polynomial (ax3 + bx2 + cx + d)
Ifα,β,andγ are the three zeroes of a cubic polynomial, then:
α+β+γ=−ab
αβ+βγ+γα=ac
αβγ=−ad
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These CBSE notes and NCERT solutions for the chapter “Polynomials”, covering topics like Zero's of Polynomials, are prepared in accordance with the latest CBSE Class 10 Maths syllabus and NCERT guidelines. The study material offers detailed explanations of important concepts, definitions, examples, and exam-orientated questions to help students understand redox reactions, corrosion, and rancidity effectively while strengthening their preparation for board examinations.
5.0Important Questions of the Zeros of Polynomials
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=515=3
Answer: The zero of the polynomial is x = 3.
Example 2: Find the zeroes of the quadratic polynomialP(x)=x2−7x+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(1×12) and add up to -7. Those numbers are -3 and -4.
x2−3x−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 α=4andβ=3.
Step 2: Verify the relationship with coefficients.
From the polynomial x2−7x+12, we have a = 1, b = -7, and c = 12.
Sum of zeroes:α+β=4+3=7. Using formula: −ab=−1−7=7 (Verified)
Product of zeroes:α×β=4×3=12 Using formula: ac=112=12. (Verified)
Example 3: Find all the zeroes of the cubic polynomial P(x)=x3−6x2+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.
Dividingx3−6x2+11x−6 by (x - 1) yields the quotient:
x2−5x+6
Step 3: Solve the resulting quadratic equation.
x2−5x+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.
6.0PREVIOUS YEAR QUESTIONS (PYQs)
Q1. Find the zeroes of the quadratic polynomial:
p(x) = x² – 5x + 6
Solution:
Factorising:
x² – 5x + 6 = x² – 2x – 3x + 6
= x(x – 2) – 3(x – 2)
= (x – 2)(x – 3)
Therefore,
x – 2 = 0 → x = 2
x – 3 = 0 → x = 3
Answer: Zeroes are 2 and 3
Q2. Find the zeroes of the polynomial: p(x) = 2x² + 7x + 3 and verify the relationship between zeroes and coefficients.