What are zeroes of a polynomial?
A zero (also called a root) of a polynomial is a number that makes the whole expression equal to zero. Imagine a seesaw: when the weight you place on one side makes the seesaw perfectly balanced, that weight is the zero.
How zeroes relate to coefficients (Vieta’s formulas)
A coefficient is the number that sits in front of each power of the variable x. Vieta’s formulas are simple shortcuts that connect the sum and product of the zeroes to those coefficients. For a quadratic ax² + bx + c, the sum of the zeroes equals -b/a and the product equals c/a. No fancy algebra, just a handy pattern.
Worked example: Quadratic polynomial
Consider 2x² - 5x + 3 = 0. Factorising gives (2x - 3)(x - 1) = 0, so the zeroes are x = 3/2 and x = 1.
- Sum of zeroes = 3/2 + 1 = 5/2 = 2.5
- Using Vieta: -b/a = -(-5)/2 = 5/2 = 2.5
- Product of zeroes = (3/2)·1 = 3/2 = 1.5
- Using Vieta: c/a = 3/2 = 1.5
Both methods give the same answer, confirming the relationship.
Worked example: Cubic polynomial
Take x³ - 6x² + 11x - 6 = 0. By trial or by factor theorem you can write it as (x-1)(x-2)(x-3) = 0. The zeroes are 1, 2 and 3.
- Sum of zeroes = 1 + 2 + 3 = 6
- Vieta for a cubic ax³ + bx² + cx + d says sum = -b/a. Here -b/a = -(-6)/1 = 6.
- Product of zeroes (taken three at a time) = 1·2·3 = 6
- Vieta says product = -d/a = -(-6)/1 = 6.
Again the formulas line up perfectly.
Quick summary table
| Degree | General form | Sum of zeroes | Product of zeroes |
|---|---|---|---|
| Quadratic | ax² + bx + c | -b/a | c/a |
| Cubic | ax³ + bx² + cx + d | -b/a | -d/a (product of all three) |
| Quartic (4th degree) | ax⁴ + bx³ + cx² + dx + e | -b/a | e/a (product of all four) |
Why this matters for CBSE Class 10
The board loves to test whether you can jump from the list of zeroes to the coefficients without doing full factorisation. Mastering Vieta’s formulas saves time and reduces mistakes in the 10‑mark questions.
💡 In Simple Words: The zeroes of a polynomial are the numbers that turn the equation into zero. Their sum and product are directly linked to the numbers sitting in front of x, x², etc. (the coefficients). Knowing the link lets you find one set when you know the other.
📝 Likely Exam Questions
- Find the sum and product of the zeroes of 3x² - 12x + 9 without solving the equation.
Answer: Sum = -(-12)/3 = 4, Product = 9/3 = 3. - If the zeroes of the cubic x³ + px² + qx + r are 2, -1 and 3, find the values of p, q and r.
Answer: Sum = -(p) = 2 + (-1) + 3 = 4 ⇒ p = -4.
Sum of pairwise products = q = (2·-1)+(2·3)+(-1·3)= -2+6-3 = 1 ⇒ q = 1.
Product = -r = 2·(-1)·3 = -6 ⇒ r = 6. - Show that for the quadratic 5x² + kx + 20, the product of its zeroes is 4. Find k.
Answer: Product = c/a = 20/5 = 4 (already satisfied). No condition on k, so any real k works. - Given that the zeroes of ax³ + bx² + cx + d are in arithmetic progression and the sum of zeroes is 9, express b in terms of a.
Answer: Let zeroes be (r‑d), r, (r+d). Sum = 3r = 9 ⇒ r = 3. By Vieta, -b/a = sum = 9 ⇒ b = -9a. - For the polynomial x⁴ - 5x³ + 8x² - 5x + 1, verify that the product of its zeroes equals the constant term (with sign considered).
Answer: Degree 4 ⇒ product = e/a = 1/1 = 1. Hence product of all four zeroes is 1, confirming Vieta’s rule.