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Class 10 Polynomials: cheat sheet
CBSE Mathematics. The zeroes of a polynomial are where its graph meets the x-axis, and for a quadratic they are tied to the coefficients by two simple rules.
Key ideas
- A zero of p(x) is a number k with p(k) = 0. Graphically, it is where y = p(x) meets the x-axis.
- A polynomial of degree n has at most n zeroes: a linear polynomial one, a quadratic two, a cubic three.
- The graph of a quadratic is a parabola: it opens upwards when a > 0 and downwards when a < 0.
- For ax² + bx + c with zeroes α and β: α + β = −b/a and αβ = c/a.
- A quadratic with a given sum and product of zeroes is k[x² − (sum)x + (product)] for any k ≠ 0.
- For a cubic ax³ + bx² + cx + d: the sum of the zeroes is −b/a, the sum of products in pairs is c/a, and the product is −d/a.
- Expressions like α² + β² or 1/α + 1/β can be found from the sum and product without finding α and β.
Remember
| Sum of zeroes (quadratic) | α + β = −b/a |
| Product of zeroes (quadratic) | αβ = c/a |
| Quadratic from its zeroes | k[x² − (α + β)x + αβ] |
| Useful identity | α² + β² = (α + β)² − 2αβ |
Watch out
- Writing the sum of zeroes as b/a instead of −b/a.
- Forgetting to clear fractions, or giving only one of the possible polynomials without the factor k.
- Reading the zeroes off the y-axis instead of the x-axis.
Worked example
Find the zeroes of 6y² − 25y + 25 and verify the relationship between the zeroes and the coefficients.
Working: 6y² − 25y + 25 = (2y − 5)(3y − 5); sum = 25/6 = −b/a, product = 25/6 = c/a
Answer: 5/3 and 5/2
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