Formula Proof and Solved Examples of Sum and Product of Zeros
Understanding Sum And Product Of Zeros In Quadratic Polynomial is crucial for exams and problem-solving in Algebra. These concepts link the solutions (roots) of a quadratic to its coefficients directly, making it easier to solve equations, factorize expressions, and even build new polynomials for both school and competitive math.
Formula Used in Sum And Product Of Zeros In Quadratic Polynomial
The standard formula is: If the quadratic polynomial is \( ax^2 + bx + c \), and its zeros are \( \alpha \) and \( \beta \), then:
Sum of zeros: \( \alpha + \beta = \frac{-b}{a} \)
Product of zeros: \( \alpha \times \beta = \frac{c}{a} \)
Here’s a helpful table to understand Sum And Product Of Zeros In Quadratic Polynomial more clearly:
Sum And Product Of Zeros In Quadratic Polynomial Table
| Expression | Formula | Description |
|---|---|---|
| Sum of Zeros | \( -\frac{b}{a} \) | Negative of (coefficient of x) divided by coefficient of \( x^2 \) |
| Product of Zeros | \( \frac{c}{a} \) | (Constant term) divided by coefficient of \( x^2 \) |
This table shows how the pattern of Sum And Product Of Zeros In Quadratic Polynomial appears regularly in real math cases.
Worked Example – Solving a Problem
1. Write the quadratic polynomial: \( p(x) = x^2 - 3x + 2 \ )2. Identify coefficients: \( a = 1 \), \( b = -3 \), \( c = 2 \ )
3. Find the sum of zeros using the formula \( -b/a \ ):
4. Find the product of zeros using the formula \( c/a \ ):
5. Factorise the polynomial: \( x^2 - 3x + 2 = (x - 1)(x - 2) \ )
6. The zeros are 1 and 2.
7. Check: \( 1 + 2 = 3 \) (matches sum), \( 1 \times 2 = 2 \) (matches product).
Final Answer: Sum = 3, Product = 2.
Practice Problems
- Find the sum and product of zeros of \( p(x) = 2x^2 + 5x + 3 \).
- Given zeros add up to 7 and product is 10, write the quadratic polynomial.
- Check if numbers -4 and 2 are zeros of \( x^2 + 2x - 8 \).
- Write the polynomial whose zeros are 5 and -6.
Common Mistakes to Avoid
- Confusing Sum And Product Of Zeros In Quadratic Polynomial with the values of the zeros themselves (they are relationships, not the roots).
- Forgetting to divide by the coefficient of \( x^2 \), not just taking values as they appear.
- Using wrong signs for the sum (remember it’s minus b).
- Mixing up the order: sum relates to b, product to c.
Real-World Applications
The concept of Sum And Product Of Zeros In Quadratic Polynomial is used in physics (projectiles), economics (profit curves), and engineering (designing parabolas). Vedantu lessons help students link such core formulas to real-world problem-solving.
We explored the idea of Sum And Product Of Zeros In Quadratic Polynomial, how to apply it, solve related problems, and understand its real-life relevance. Practice more with Vedantu to build confidence in these concepts, and for more details on roots and factorization, see Roots of Polynomial Equation and Factor Theorem.
FAQs on Sum and Product of Zeros of a Quadratic Polynomial Explained
1. What is the sum and product of zeros in a quadratic polynomial?
The sum of zeros of a quadratic polynomial ax² + bx + c is -b/a and the product of zeros is c/a, where a ≠ 0.
For a quadratic polynomial of the form ax² + bx + c:
- If the zeros are α and β, then α + β = -b/a
- αβ = c/a
2. How do you find the sum and product of zeros of a quadratic equation?
To find the sum and product of zeros, compare the quadratic equation with the standard form ax² + bx + c.
Steps:
- Write the equation in the form ax² + bx + c = 0
- Identify values of a, b, and c
- Use the formulas: Sum = -b/a, Product = c/a
- a = 2, b = 5, c = 3
- Sum = -5/2
- Product = 3/2
3. What is the formula for the sum of zeros of a quadratic polynomial?
The formula for the sum of zeros of a quadratic polynomial ax² + bx + c is -b/a.
If the zeros are α and β, then:
- α + β = -b/a
4. What is the formula for the product of zeros of a quadratic polynomial?
The formula for the product of zeros of a quadratic polynomial ax² + bx + c is c/a.
If the zeros are α and β, then:
- αβ = c/a
5. How do you verify the sum and product of zeros with an example?
You can verify the formulas by finding the zeros and checking whether their sum and product match -b/a and c/a.
Example: For x² − 7x + 10 = 0:
- Factor: (x − 5)(x − 2) = 0
- Zeros are 5 and 2
- Sum = 5 + 2 = 7
- Product = 5 × 2 = 10
- -b/a = -(-7)/1 = 7
- c/a = 10/1 = 10
6. Can you form a quadratic polynomial if the sum and product of zeros are given?
Yes, a quadratic polynomial can be formed using the identity x² − (sum)x + product.
If sum = S and product = P, then the polynomial is:
- x² − Sx + P
- Polynomial = x² − 8x + 15
7. Why is the sum of zeros equal to -b/a in a quadratic polynomial?
The sum of zeros equals -b/a because of the relationship derived from the quadratic formula.
For ax² + bx + c = 0, the zeros are:
- (-b ± √(b² − 4ac)) / 2a
- α + β = -b/a
8. What happens to the sum and product of zeros if the leading coefficient is 1?
If the leading coefficient a = 1, then the sum of zeros is -b and the product is c.
For a quadratic polynomial x² + bx + c:
- Sum = -b
- Product = c
9. What is the relationship between zeros and coefficients of a quadratic polynomial?
The zeros of a quadratic polynomial are directly related to its coefficients through -b/a and c/a.
For ax² + bx + c with zeros α and β:
- α + β = -b/a
- αβ = c/a
10. What are common mistakes when finding the sum and product of zeros?
Common mistakes include using the wrong sign for b or forgetting to divide by a.
Frequent errors:
- Writing sum as b/a instead of -b/a
- Forgetting that the equation must be in the form ax² + bx + c = 0
- Ignoring the condition a ≠ 0
