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What is the Zeroes of a Polynomial?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
The 'zeroes' of a polynomial are the specific values of the variable (like 'x') that make the entire polynomial equal to zero. Think of them as the 'roots' or 'solutions' to the polynomial equation when it's set to zero.
Simple Example
Quick Example
Imagine you have a polynomial P(x) = x - 5. If you want to find its zero, you ask: 'What value of x will make x - 5 equal to 0?' The answer is x = 5. So, 5 is the zero of this polynomial.
Worked Example
Step-by-Step
Let's find the zeroes of the polynomial P(x) = x^2 - 4x + 3.
STEP 1: Set the polynomial equal to zero: x^2 - 4x + 3 = 0.
---STEP 2: Factor the quadratic expression. We need two numbers that multiply to 3 and add up to -4. These numbers are -1 and -3.
---STEP 3: Rewrite the equation using the factors: (x - 1)(x - 3) = 0.
---STEP 4: For the product of two terms to be zero, at least one of the terms must be zero.
---STEP 5: Set each factor equal to zero: x - 1 = 0 OR x - 3 = 0.
---STEP 6: Solve each simple equation: x = 1 OR x = 3.
---ANSWER: The zeroes of the polynomial P(x) = x^2 - 4x + 3 are 1 and 3.
Why It Matters
Common Mistakes
MISTAKE: Confusing the constant term with a zero | CORRECTION: The zero is the value of 'x' that makes the polynomial zero, not necessarily the constant term itself.
MISTAKE: Forgetting to set the polynomial to zero before solving | CORRECTION: To find zeroes, you MUST always set P(x) = 0 first, otherwise you are just evaluating the polynomial.
MISTAKE: Assuming all polynomials have only one zero | CORRECTION: A polynomial can have multiple zeroes. For example, a quadratic polynomial (degree 2) can have up to two zeroes.
Practice Questions
Try It Yourself
QUESTION: Find the zero of the polynomial P(x) = 2x - 10. | ANSWER: 5
QUESTION: Find the zeroes of the polynomial Q(x) = x^2 - 9. | ANSWER: 3 and -3
QUESTION: If P(x) = x^2 - 5x + k and one of its zeroes is 2, find the value of k. | ANSWER: 6
MCQ
Quick Quiz
Which of the following is a zero of the polynomial P(x) = 3x + 6?
6
3
-2
The Correct Answer Is:
C
To find the zero, set 3x + 6 = 0. Subtract 6 from both sides to get 3x = -6. Divide by 3 to get x = -2. So, -2 is the zero.
Real World Connection
In the Real World
Imagine an ISRO scientist launching a rocket. The path of the rocket can be described by a polynomial equation. Finding the zeroes of this polynomial helps determine when the rocket will hit a certain altitude (like the ground, if it's a test launch) or reach its peak, crucial for mission planning.
Key Vocabulary
Key Terms
POLYNOMIAL: An expression with variables and coefficients, involving only operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. | VARIABLE: A symbol (like x or y) that represents a value that can change. | COEFFICIENT: A number multiplied by a variable in an algebraic expression. | DEGREE OF POLYNOMIAL: The highest exponent of the variable in a polynomial.
What's Next
What to Learn Next
Great job understanding zeroes! Next, you should explore the 'Relationship between Zeroes and Coefficients of a Polynomial'. This will show you how the zeroes are directly linked to the numbers in the polynomial, opening up new ways to solve problems!
