Question

if f(1) = 0, what are all the roots of the function f(x)=x³+3x²−x−3? use the remainder theorem.\
\\(\bigcirc\\) x = −1, x = 1, or x = 3\
\\(\bigcirc\\) x = −3, x = −1, or x = 1\
\\(\bigcirc\\) x = −3 or x = 1\
\\(\bigcirc\\) x = −1 or x = 3

Explanation:

Step1: Factor the polynomial

Given \( f(x) = x^3 + 3x^2 - x - 3 \), we can factor by grouping.
Group the terms: \( (x^3 + 3x^2) + (-x - 3) \)
Factor out common factors: \( x^2(x + 3) - 1(x + 3) \)
Now, factor out \( (x + 3) \): \( (x + 3)(x^2 - 1) \)
We know that \( x^2 - 1 \) is a difference of squares, so it factors further: \( (x + 3)(x - 1)(x + 1) \)

Step2: Find the roots

To find the roots, set \( f(x) = 0 \):
\( (x + 3)(x - 1)(x + 1) = 0 \)
Using the zero - product property, if \( ab = 0 \), then either \( a = 0 \), \( b = 0 \) or both.
So, \( x+3 = 0\) gives \( x=-3 \); \( x - 1=0\) gives \( x = 1\); \( x + 1=0\) gives \( x=-1 \)

Answer:

\( x=-3 \), \( x = - 1 \), or \( x = 1 \) (corresponding to the option: \( x=-3,x = - 1,\text{or }x = 1 \))