Which of the following statements is correct?

Class 9PolynomialsPolynomials

AThe degree of the zero polynomial is 0.
BA polynomial can have a negative exponent.
CThe remainder on dividing p $(x)$ by $(x - a)$ is p(a).
DEvery quadratic polynomial has exactly two distinct real zeros.

Step-by-Step Solution

The Remainder Theorem is the correct statement. The zero polynomial's degree is undefined; polynomials forbid negative exponents; some quadratics (like

x^{2} + 1

) have no real zeros.

Answer: C) The remainder on dividing p

(x)

(x - a)

is p(a).

Key idea: Remainder Theorem

**When

p (x)

is divided by

(x - a)

, the remainder =

p

(a).**

Don't do long division — just substitute

x = a

into the polynomial.

Example: divide

p (x) = x^{3} - 2 x + 1

(x - 2)

. Remainder =

p (2) = 8 - 4 + 1 = 5

.

Why it works: any polynomial can be written as

p (x) = q (x) \times (x - a) + r

, where

r

is a constant remainder. Substituting

x = a

kills the

q (x) \times (x - a)

term, leaving

p (a) = r

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