Step-by-Step Method to Calculate 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 How to Find Quadratic Polynomial Using Sum and Product of Zeros
1. How do you find the quadratic polynomial if the sum and product of its zeros are given?
To find a quadratic polynomial when the sum (S) and product (P) of its zeros are given, use the general form: x2 - (sum of zeros)x + (product of zeros). Therefore, the required quadratic polynomial is x2 - Sx + P, where S is the sum and P is the product of zeros.
2. What if the sum and product of the zeros of a polynomial are 3 and -10 respectively?
If the sum of zeros is 3 and the product of zeros is -10, the quadratic polynomial can be written as x2 - 3x - 10.
3. What is the sum and product rule of a quadratic equation?
The sum and product rule says that for a quadratic equation ax2 + bx + c = 0:
• Sum of zeros = -b/a
• Product of zeros = c/a
4. What is the sum and product of zeros of the quadratic polynomial x2 + 15?
For the quadratic polynomial x2 + 15, the sum of zeros is 0 and the product of zeros is 15.
5. Find the sum and product of zeros of quadratic polynomial x2 + 3.
In x2 + 3, the coefficient of x is 0 and the constant term is 3. So, sum of zeros = 0, and product of zeros = 3.
6. The sum and product of zeros of the quadratic polynomial 2x2 - 7x + 5 are respectively?
For 2x2 - 7x + 5, use the formulas:
• Sum of zeros = -(-7)/2 = 7/2
• Product of zeros = 5/2
7. Find the sum and product of zeros of the quadratic polynomial 5 - 3x2 + 7x.
First arrange as –3x2 + 7x + 5.
• Sum of zeros = –b/a = –7/(–3) = 7/3
• Product of zeros = c/a = 5/(–3) = –5/3
8. What is the formula for sum and product of zeros of a quadratic polynomial?
For ax2 + bx + c:
• Sum of zeros = –b/a
• Product of zeros = c/a
9. What are the steps to find zeros of a quadratic polynomial?
To find zeros of a quadratic polynomial ax2 + bx + c:
1. Set the expression equal to zero.
2. Factorise or use the quadratic formula: x = [–b ± √(b2 – 4ac)]/(2a).
3. Solve for the values of x (the zeros).
10. What are the zeroes of the quadratic polynomial x2 + 99x + 127?
The zeros can be found by solving x2 + 99x + 127 = 0 using the quadratic formula:
x = [–99 ± √(992 – 4 × 1 × 127)]/2
Calculate inside the square root, solve for both x values.
11. Can you generate a quadratic polynomial when sum and product of zeros are given as variables?
Yes. If sum of zeros = S and product of zeros = P, the quadratic polynomial is x2 – Sx + P.
12. How are sum and product of roots used in CBSE exams?
In CBSE exams, questions may ask you to find or construct quadratic polynomials using sum and product of zeros, calculate sum and product from given expressions, or verify if given numbers are zeros. Knowing the formulas and steps helps solve such problems quickly and correctly.
