When p $(x)$ is divided by $(x - 3)$ the remainder is 0. Which conclusion must be true?

Class 9PolynomialsPolynomials

Step-by-Step Solution

Remainder 0 means

p (3) = 0

, and by the Factor Theorem that makes

(x - 3)

a factor — the factor is

(x

− zero), so it is

(x - 3)

, not

(x + 3)

. Nothing follows about the individual coefficients.

Answer: D)

(x - 3)

is a factor of p

(x)

(x - a)

is a factor of

p (x)

if and only if

p (a) = 0

.**

It's a special case of the remainder theorem (with remainder 0).

Use it to test factors:
Is

(x + 3)

a factor of

p (x) = x^{2} + 5 x + 6

?
Substitute

x = - 3

(note:

x + 3 = 0

→

x = - 3) . p (- 3) = 9 - 15 + 6 = 0

. ✓ Yes,

(x + 3)

is a factor.

This is the standard way to factor cubics — guess and test small zeros.

Don't just memorise the answer — build a daily practice habit.