What is the zero of a polynomial?

A zero (root) of a polynomial p(x) is a value of x for which p(x) = 0. Geometrically, zeroes are the x-intercepts of the graph. A degree-n polynomial has at most n real zeroes. Example: p(x) = x² − 5 has zeroes at x = ±√5 because p(√5) = 5 − 5 = 0.

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

For p(x) = ax² + bx + c with zeroes α and β: Sum of zeroes = α + β = −b/a, Product of zeroes = α × β = c/a. Example: p(x) = 2x² − 7x + 3: α + β = 7/2, αβ = 3/2. These relationships allow us to find a quadratic given its zeroes: p(x) = k[x² − (sum)x + (product)].

What is the Factor Theorem?

Factor Theorem: (x − a) is a factor of p(x) if and only if p(a) = 0. It is a special case of the Remainder Theorem (when remainder = 0). Use it to check if a linear expression is a factor — substitute x = a into p(x). If result is 0, (x − a) is a factor and you can divide to find remaining factors.

Zeroes of Polynomials

Polynomials • Class 9 Mathematics • NCERT • CBSE

A zero (root) of a polynomial p(x) is a value of x for which p(x) = 0. A polynomial of degree n has at most n zeroes. For a quadratic ax²+bx+c: sum of zeroes = −b/a, product of zeroes = c/a.

Key Formulas

Sum of zeroes of ax² + bx + c: α + β = −b/a
Product of zeroes of ax² + bx + c: αβ = c/a
(x + y)² = x² + 2xy + y²
(x − y)² = x² − 2xy + y²
x³ + y³ + z³ − 3xyz = (x+y+z)(x²+y²+z²−xy−yz−zx)

Frequently Asked Questions

What is the zero of a polynomial?: A zero (root) of a polynomial p(x) is a value of x for which p(x) = 0. Geometrically, zeroes are the x-intercepts of the graph. A degree-n polynomial has at most n real zeroes. Example: p(x) = x² − 5 has zeroes at x = ±√5 because p(√5) = 5 − 5 = 0.
What is the relationship between zeroes and coefficients of a quadratic?: For p(x) = ax² + bx + c with zeroes α and β: Sum of zeroes = α + β = −b/a, Product of zeroes = α × β = c/a. Example: p(x) = 2x² − 7x + 3: α + β = 7/2, αβ = 3/2. These relationships allow us to find a quadratic given its zeroes: p(x) = k[x² − (sum)x + (product)].
What is the Factor Theorem?: Factor Theorem: (x − a) is a factor of p(x) if and only if p(a) = 0. It is a special case of the Remainder Theorem (when remainder = 0). Use it to check if a linear expression is a factor — substitute x = a into p(x). If result is 0, (x − a) is a factor and you can divide to find remaining factors.

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