Question

One factor of x^3 + 3x^2 - x - 3. OPTIONS:
(A) x + 1
(B) x + 2
(C) x - 2
(D) x - 3

Answer 1

Step-by-step explanation:

Given -

• p(x) = x³ + 3x² - x - 3

Here we check all the options then we find which option is correct.

Now,

Option 1 :-

If x + 1 is a factor of polynomial then -1 is zero of that polynomial and after substituting x = -1 if it comes equal to zero (0) then we say that this option is correct.

→ p(x) = x³ + 3x² - x - 3

→ p(-1) = (-1)³ + 3(1)² - (-1) - 3

→ -1 + 3 + 1 - 3

→ 0

Here, after substituting x = -1 it comes equal to zero and we say that x + 1 is a factor of the polynomial.

Option 2 :-

If x + 2 is a factor of polynomial then -2 is zero of that polynomial and after substituting x = -2 if it comes equal to zero (0) then we say that this option is correct.

→ p(x) = x³ + 3x² - x - 3

→ p(-2) = (-2)³ + 3(2)² - (-2) - 3

→ -8 + 12 + 2 - 3

→ 3

Here, after substituting x = -2 it doesn't comes equal to zero and we say that x + 2 isn't a factor of the polynomial.

Option 3 :-

If x - 2 is a factor of polynomial then 2 is zero of that polynomial and after substituting x = 2 if it comes equal to zero (0) then we say that this option is correct.

→ p(x) = x³ + 3x² - x - 3

→ p(2) = (2)³ + 3(2)² - (2) - 3

→ 8 + 12 - 2 - 3

→ 15

Here, after substituting x = 2 it doesn't comes equal to zero and we say that x - 2 isn't a factor of the polynomial.

Option 4 :-

If x - 3 is a factor of polynomial then 3 is zero of that polynomial and after substituting x = 3 if it comes equal to zero (0) then we say that this option is correct.

→ p(x) = x³ + 3x² - x - 3

→ p(3) = (3)³ + 3(3)² - (3) - 3

→ 27 + 27 - 3 - 3

→ 48

Here, after substituting x = 3 it doesn't comes equal to zero and we say that x - 3 isn't a factor of the polynomial.

From above discussion we concluded that x + 1 is factor of x³ + 3x² - x - 3

Hence,

Option 1 is correct.

Answer 2

✯✯ QUESTION ✯✯

One factor of x^3 + 3x^2 - x - 3.

(A) x + 1

(B) x + 2

(C) x - 2

(D) x - 3

━━━━━━━━━━━━━━━━━━━━

✰✰ ANSWER ✰✰

A)$⇢\textbf{x+1}$

$⇢\textbf{x=-1}$

$⇢{x}^{3}+3{x}^{2}-x-3$

$⇢{(-1)}^{3}+3{(-1)}^{2}-(-1)-3$

$⇢-1+3+1-3$

$⇢0$

So,x+1 is a factor of ${x}^{3}+3{x}^{2}-x-3$.

_______________________

B) $⇢\textbf{x+2}$

$⇢\textbf{x=-2}$

$⇢{x}^{3}+3{x}^{2}-x-3$

$⇢{(-2)}^{3}+3{(-2)}^{2}-(-2)-3$

$⇢-8+12+2-3$

$⇢4+2-3$

$⇢3$

So,x+2 is not a factor of ${x}^{3}+3{x}^{2}-x-3$

_______________________

C) $⇢\textbf{x-2}$

$⇢\textbf{x=2}$

$⇢{x}^{3}+3{x}^{2}-x-3$

$⇢{(2)}^{3}+{(3)}^{2}-(2)-3$

$⇢8+9-2-3$

$⇢17-2-3$

$⇢12$

So,x-2 is not a factor of ${x}^{3}+3{x}^{2}-x-3$

_______________________

D) $⇢\textbf{x-3}$

$⇢\textbf{x=3}$

$⇢{x}^{3}+3{x}^{2}-x-3$

$⇢{(3)}^{3}+{(3)}^{2}-(2)-3$

$⇢27+9-2-3$

$⇢31$

So,x-3 is not a factor of ${x}^{3}+3{x}^{2}-x-3$

➜Option A is correct....