Question
If α, β, and γ are the zeroes of the polynomial p(x) = 2x³ - 5x² + 4x - 3, find the values of:
(i) α + β + γ (ii) αβ + βγ + γα (iii) αβγ
This is a direct CBSE 2024 Board Exam question — knowing these three formulas cold will fetch you full marks in under 90 seconds.
Solution — Step by Step
Write the polynomial in standard form ax³ + bx² + cx + d and read off each coefficient carefully.
For p(x) = 2x³ - 5x² + 4x - 3, we have:
The sum of all three zeroes comes from the relationship between the x² term and the leading coefficient.
Take zeroes two at a time — this gives us the coefficient of x divided by the leading coefficient.
The product of all three zeroes connects to the constant term — and watch the sign here.
Final Answers:
Why This Works
These formulas come directly from how polynomials factor. If α, β, γ are the three zeroes, we can write p(x) = a(x - α)(x - β)(x - γ). Expanding this product gives us a[x³ - (α+β+γ)x² + (αβ+βγ+γα)x - αβγ].
Now compare this with ax³ + bx² + cx + d. Match the coefficient of x² on both sides: a·(-(α+β+γ)) = b, which gives α+β+γ = -b/a. The other two formulas follow the same logic from the x term and the constant.
This is Vieta’s Formulas — a powerful idea that lets us find sums and products of roots without actually finding the roots themselves. CBSE loves asking this because it tests whether you understand the structure of polynomials, not just the quadratic formula.
Alternative Method
For practice, verify by checking dimensions. Suppose we already know one zero — say α = 1 (test: 2(1) - 5(1) + 4(1) - 3 = -2, not zero, so 1 isn’t a root here). The formula method is always faster for board exams than actually solving the cubic.
Board exam shortcut: Write all four values a, b, c, d in a row first — students who skip this step often mix up signs halfway through. The formula for product of zeroes is -d/a, not d/a. One second of writing saves a silly half-mark error.
If a question gives you the zeroes instead and asks for the polynomial, just reverse the process: p(x) = a[x³ - (sum)x² + (sum of pairs)x - (product)]. This reverse direction appeared in CBSE 2023.
Common Mistake
The sign trap on αβγ. Students write αβγ = d/a (forgetting the negative sign). The correct formula is αβγ = -d/a. In our problem, d = -3, so -d/a = -(-3)/2 = 3/2. If you wrote d/a, you’d get -3/2 — wrong sign, lost mark. The negative is baked into the formula, not the value of d.
A quick sanity check: for x³ - 6x² + 11x - 6, the zeroes are 1, 2, 3. Their product is 6. Here d = -6 and a = 1, so -d/a = 6. Confirmed. Use this as your mental anchor whenever you’re unsure of the sign.