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What is the Zeroes of a Polynomial by Factorisation?
Grade Level:
Class 7
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The 'zeroes' of a polynomial are the values of the variable that make the entire polynomial equal to zero. When we find these values by 'factorisation', it means we break down the polynomial into simpler parts (factors) and then set each factor to zero to find the solutions.
Simple Example
Quick Example
Imagine you have a polynomial like (x - 2)(x - 3). If you want to find the zeroes, you need to find the 'x' values that make this expression zero. Just like if you multiply two numbers and the answer is zero, one of the numbers must be zero. So, either (x - 2) is zero or (x - 3) is zero.
Worked Example
Step-by-Step
Let's find the zeroes of the polynomial x^2 - 5x + 6 by factorisation.
1. First, we need to factorise the quadratic expression x^2 - 5x + 6.
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2. We look for two numbers that multiply to +6 and add up to -5. These numbers are -2 and -3.
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3. So, we can rewrite the polynomial as (x - 2)(x - 3).
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4. Now, to find the zeroes, we set the entire polynomial equal to zero: (x - 2)(x - 3) = 0.
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5. For the product of two factors to be zero, at least one of the factors must be zero. So, we set each factor to zero separately.
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6. Case 1: x - 2 = 0. Adding 2 to both sides gives x = 2.
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7. Case 2: x - 3 = 0. Adding 3 to both sides gives x = 3.
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8. The zeroes of the polynomial x^2 - 5x + 6 are 2 and 3.
Why It Matters
Understanding zeroes helps us solve many real-world problems, from designing flight paths for ISRO rockets to predicting stock market trends. Engineers use it to find optimal designs, and data scientists use it to understand data patterns. It's a key skill for careers in technology and research.
Common Mistakes
MISTAKE: Not setting the factors equal to zero after factorising. Students might factorise correctly but forget the final step of finding 'x'. | CORRECTION: After factorising, remember that for the product to be zero, each factor must be set to zero individually to find the possible values of the variable.
MISTAKE: Incorrectly factorising the polynomial. Forgetting rules for signs or finding wrong pairs of numbers. | CORRECTION: Double-check your factorisation by multiplying the factors back together to ensure you get the original polynomial.
MISTAKE: Only finding one zero when there should be two (or more). Forgetting that a quadratic polynomial usually has two zeroes. | CORRECTION: Always check how many factors you have. If you have two distinct factors, you will usually have two distinct zeroes.
Practice Questions
Try It Yourself
QUESTION: Find the zeroes of the polynomial (x - 5)(x + 1) by factorisation. | ANSWER: 5, -1
QUESTION: Find the zeroes of the polynomial x^2 - 7x + 10 by factorisation. | ANSWER: 2, 5
QUESTION: Find the zeroes of the polynomial 2x^2 + 5x - 3 by factorisation. | ANSWER: 1/2, -3
MCQ
Quick Quiz
What are the zeroes of the polynomial (x + 4)(x - 7)?
4 and 7
-4 and -7
-4 and 7
4 and -7
The Correct Answer Is:
C
For (x + 4)(x - 7) = 0, either x + 4 = 0 (which means x = -4) or x - 7 = 0 (which means x = 7). So the zeroes are -4 and 7.
Real World Connection
In the Real World
Imagine a drone delivering packages for Zepto. Its flight path can sometimes be described by polynomial equations. Finding the zeroes helps determine when the drone will be at ground level or at a certain height, which is crucial for safe navigation and delivery planning.
Key Vocabulary
Key Terms
POLYNOMIAL: An expression with variables and coefficients, involving only operations of addition, subtraction, multiplication, and non-negative integer exponents. | ZEROES: The values of the variable that make a polynomial equal to zero. | FACTORISATION: The process of breaking down a polynomial into a product of simpler polynomials (factors). | QUADRATIC: A polynomial of degree 2 (highest power of the variable is 2).
What's Next
What to Learn Next
Great job learning about zeroes by factorisation! Next, you can explore 'Quadratic Formula' to find zeroes of polynomials that are harder to factorise. This will give you another powerful tool for solving even more complex problems!
