For $p (x) = (x - 1) (x + 2) (x - 5)$ , which statement is true?

Class 9PolynomialsPolynomials

Step-by-Step Solution

A product is zero exactly when one factor is zero.

(x + 2) = 0

x = - 2

, so

p (- 2) = 0

. The zeros are 1, −2 and 5. Checking the others:

p (2) = (1) (4) (- 3) = - 12, p (0) = (- 1) (2) (- 5) = 10

, and

p (1) = 0

so '

p

(1) ≠ 0' is false.

Answer: B)

p (- 2) = 0

(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.

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