How to Form a Quadratic Polynomial from the Sum and Product of Its Zeroes (Class 10)
In one line: to form a quadratic polynomial from the sum and product of its zeroes, use
x² − (sum of zeroes)x + (product of zeroes).
If the sum of the zeroes is S and the product is P, the quadratic polynomial is simply x² − Sx + P. For example, if the zeroes add up to 5 and multiply to 6, the polynomial is x² − 5x + 6. This is the reverse of the previous topic: instead of starting from a polynomial and finding the sum and product, you start from the sum and product and build the polynomial.
This guide covers that skill for CBSE Class 10 (Section 2.3, Example 4 and Exercise 2.2). It builds directly on the relationship between zeroes and coefficients, where you learned that sum of zeroes = −b/a and product of zeroes = c/a. Here we run those same two facts backwards.
▶️ Watch the 30-second reel: Can you form the polynomial? ✍️
The formula (and why it works)
To build a quadratic polynomial whose zeroes have sum S and product P:
Quadratic polynomial = x² − Sx + P = x² − (sum of zeroes)x + (product of zeroes)
Why this works. From the relationship between zeroes and coefficients, a quadratic with zeroes α and β can be written as
x² − (α + β)x + αβ.
Since α + β is the sum and αβ is the product, replacing them with S and P gives x² − Sx + P. So the middle coefficient is the negative of the sum, and the constant term is the product.
The one sign to watch: the sum comes in with a minus in front (−Sx), while the product is just added (+P). Get that minus right and the rest is substitution.
▶️ Watch the 30-second reel: One minus sign changes everything! 👀
Worked example 1 — sum and product given directly
Find a quadratic polynomial whose zeroes have sum −3 and product 2.
Here S = −3 and P = 2. Substitute into x² − Sx + P:
x² − (−3)x + 2 = x² + 3x + 2
Notice the double negative: −(−3)x becomes +3x. Quick check: x² + 3x + 2 = (x + 1)(x + 2), so the zeroes are −1 and −2. Their sum is −3 ✓ and product is 2 ✓ — exactly what we were given.
Worked example 2 — clearing fractions with k
Find a quadratic polynomial whose zeroes have sum ¼ and product −1.
Substitute S = ¼ and P = −1:
x² − ¼x + (−1) = x² − ¼x − 1
This is already a correct answer, but board answers are usually written with whole-number coefficients. Because any multiple of a polynomial has the same zeroes, we can multiply the whole thing by a convenient number — here 4, to clear the quarter:
4(x² − ¼x − 1) = 4x² − x − 4
Quick check using the relationship from the previous topic: for 4x² − x − 4, sum = −b/a = −(−1)/4 = ¼ ✓ and product = c/a = −4/4 = −1 ✓. Choosing the right multiplier to clear denominators is a small but frequently tested move — drill a mixed set of fractional and surd cases in the Topic Practice on examfront, where the Practice Companion shows you the cleanest multiplier each time.
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Worked example 3 — building from the zeroes themselves
Sometimes you are given the two zeroes instead of their sum and product. The method is the same — just compute the sum and product first.
Find a quadratic polynomial whose zeroes are 2 and −5.
- Sum: S = 2 + (−5) = −3
- Product: P = 2 × (−5) = −10
Substitute into x² − Sx + P:
x² − (−3)x + (−10) = x² + 3x − 10
Quick check: x² + 3x − 10 = (x + 5)(x − 2), whose zeroes are −5 and 2 ✓.
▶️ Watch the 30-second reel: Find the polynomial from its zeroes! ✍️
Worked example 4 — when the sum is zero
If the sum of the zeroes is 0, the middle term vanishes and you are left with just x² + (product).
Find a quadratic polynomial whose zeroes are √3 and −√3.
- Sum: S = √3 + (−√3) = 0
- Product: P = (√3)(−√3) = −3
Substitute: x² − (0)x + (−3) = x² − 3. That matches the familiar polynomial x² − 3, whose zeroes are indeed √3 and −√3. So a “missing middle term” is a signal that the two zeroes are equal and opposite.
Why there is more than one answer
For a given sum S and product P, every polynomial of the form
k(x² − Sx + P), where k is any non-zero real number,
has the same zeroes — because multiplying a polynomial by a constant does not move where it equals zero. So x² − 5x + 6, 2x² − 10x + 12 and −3x² + 15x − 18 all share the zeroes 2 and 3.
In practice: if the question just says “find a quadratic polynomial,” give the simplest one with k = 1 (that is, x² − Sx + P). Only reach for a different k when you need to clear fractions and present tidy whole-number coefficients, as in Worked example 2. Recognising when a question wants the simplest answer versus whole-number coefficients is the kind of exam judgement the Mistake Identification tool on examfront helps you sharpen.
▶️ Watch the 30-second reel: Equal zeroes challenge! 🎯
We have kept to the core cases here. The tougher board variations — building polynomials from conditions on the zeroes (such as “one zero is twice the other” or “the zeroes differ by 3”), surd-heavy sums and products, and full word-problem framings — are worked through step by step inside examfront, where Progress Tracking shows you when the skill is exam-ready.
Key Takeaways
- To form a quadratic from sum S and product P of the zeroes: x² − Sx + P.
- The middle term is −(sum); the constant term is the product.
- Watch double negatives: a negative sum gives a positive middle term.
- Given the zeroes instead, first find their sum and product, then substitute.
- The answer is not unique — any k(x² − Sx + P) works; use k = 1 for the simplest, or a k that clears fractions.
- Sum = 0 removes the middle term, leaving x² + product.
Quick Facts
- Formula: quadratic polynomial = x² − (sum of zeroes)x + (product of zeroes).
- Middle coefficient: −(sum of zeroes).
- Constant term: product of zeroes.
- Family of answers: k(x² − Sx + P), k ≠ 0.
- Clearing fractions: multiply by the LCM of the denominators.
- Chapter: Polynomials · Class: 10 · Subject: Maths · Board: CBSE.
Common Mistakes
- Adding the sum instead of subtracting it. The middle term is −(sum), so the polynomial is x² − Sx + P, not x² + Sx + P. Fix: write the minus first, then insert the sum with its own sign.
- Mishandling a double negative. For sum = −3, the middle term is −(−3)x = +3x, not −3x. Fix: substitute the sum inside brackets first, then simplify the signs.
- Swapping the sum and product. Putting the product where the sum belongs (or vice-versa) gives the wrong polynomial. Fix: remember “sum sits with x, product sits alone.”
- Leaving fractions when whole numbers are expected. x² − ¼x − 1 is correct but often should be written as 4x² − x − 4. Fix: multiply by the denominator’s LCM to tidy the coefficients.
- Thinking there is only one answer. Any k(x² − Sx + P) is valid. Fix: give x² − Sx + P unless the question asks for whole-number coefficients or a specific leading coefficient.
FAQ
Q. How do you form a quadratic polynomial from the sum and product of its zeroes? Use x² − (sum)x + product. If the sum is S and the product is P, the polynomial is x² − Sx + P. For sum = 5 and product = 6, it is x² − 5x + 6.
Q. What is the formula for a quadratic polynomial when the zeroes are given? First find the sum and product of the two zeroes, then substitute into x² − (sum)x + product. For zeroes 2 and 7: sum = 9, product = 14, so the polynomial is x² − 9x + 14.
Q. Is the quadratic polynomial formed this way unique? No. Any k(x² − Sx + P) with k ≠ 0 has the same zeroes. Take k = 1 for the simplest answer; use a suitable k to clear fractions.
Q. How do you form a quadratic polynomial when the sum or product is a fraction? Write x² − Sx + P, then multiply the whole polynomial by a value that clears the denominators. For sum = ½ and product = −3, x² − ½x − 3 becomes 2x² − x − 6.
Q. What if the sum of the zeroes is 0? The middle term disappears, leaving x² + product. For zeroes √3 and −√3, the sum is 0 and the product is −3, so the polynomial is x² − 3.
Related Concepts
- Relationship Between Zeroes and Coefficients of a Polynomial: the forward direction — going from a polynomial to the sum and product of its zeroes.
- Zeroes of a Polynomial — meaning and how to find: what a zero is and how to compute it.
- Geometrical Meaning of the Zeroes of a Polynomial: how these zeroes show up on a graph.
- Quadratic Equations Class 10 — chapter guide: where building expressions from roots continues into Chapter 4.
- Polynomials Class 10 — how to approach the chapter: the full chapter roadmap and where this section fits.
Ready to make this automatic? Build a few polynomials from sums and products, then flip to verifying them, inside the Topic Practice on examfront — the Practice Companion catches sign slips, and Progress Tracking tells you when you’ve mastered it. Start on examfront →