What is the relationship between zeroes and coefficients of a quadratic polynomial?

For ax² + bx + c with zeroes α and β: Sum of zeroes = α + β = −b/a (negative of coefficient of x divided by leading coefficient). Product of zeroes = αβ = c/a (constant term divided by leading coefficient). Example: for 3x² − 5x + 2, sum = 5/3, product = 2/3. These formulas let you check zeroes without actually factorising.

How do you form a quadratic polynomial if its zeroes are given?

Use the formula: p(x) = k[x² − (sum of zeroes)x + (product of zeroes)] where k is any non-zero constant (usually k = 1). Example: zeroes are 3 and −4. Sum = 3+(−4) = −1; Product = 3×(−4) = −12. So p(x) = x² − (−1)x + (−12) = x² + x − 12. Verify: p(3) = 9+3−12=0 ✓, p(−4) = 16−4−12=0 ✓

What are the zeroes of a polynomial graphically?

The zeroes of a polynomial p(x) are the x-coordinates of points where the graph of y = p(x) crosses or touches the x-axis. For a quadratic (parabola): if it crosses x-axis at 2 points → 2 distinct real zeroes; if it just touches the x-axis → 1 repeated zero; if it doesn't touch → no real zeroes (complex roots). A cubic always crosses the x-axis at least once (at least 1 real zero).

Zeroes and Coefficients

Polynomials • Class 10 Mathematics • NCERT • CBSE

For a quadratic polynomial ax² + bx + c with zeroes α and β: Sum of zeroes = α + β = −b/a; Product of zeroes = αβ = c/a. For cubic polynomial ax³ + bx² + cx + d with zeroes α, β, γ: sum = −b/a, sum of products taken two at a time = c/a, product = −d/a.

Key Formulas

Sum of zeroes (quadratic): α + β = −b/a
Product of zeroes (quadratic): αβ = c/a
Quadratic from zeroes: x² − (α+β)x + αβ = 0
Sum of zeroes (cubic): α+β+γ = −b/a
Product of zeroes (cubic): αβγ = −d/a
Division algorithm: p(x) = g(x)·q(x) + r(x)

Frequently Asked Questions

What is the relationship between zeroes and coefficients of a quadratic polynomial?: For ax² + bx + c with zeroes α and β: Sum of zeroes = α + β = −b/a (negative of coefficient of x divided by leading coefficient). Product of zeroes = αβ = c/a (constant term divided by leading coefficient). Example: for 3x² − 5x + 2, sum = 5/3, product = 2/3. These formulas let you check zeroes without actually factorising.
How do you form a quadratic polynomial if its zeroes are given?: Use the formula: p(x) = k[x² − (sum of zeroes)x + (product of zeroes)] where k is any non-zero constant (usually k = 1). Example: zeroes are 3 and −4. Sum = 3+(−4) = −1; Product = 3×(−4) = −12. So p(x) = x² − (−1)x + (−12) = x² + x − 12. Verify: p(3) = 9+3−12=0 ✓, p(−4) = 16−4−12=0 ✓
What are the zeroes of a polynomial graphically?: The zeroes of a polynomial p(x) are the x-coordinates of points where the graph of y = p(x) crosses or touches the x-axis. For a quadratic (parabola): if it crosses x-axis at 2 points → 2 distinct real zeroes; if it just touches the x-axis → 1 repeated zero; if it doesn't touch → no real zeroes (complex roots). A cubic always crosses the x-axis at least once (at least 1 real zero).

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