How many times does the graph of `y=x^(3)-3x^(2)-x+3` intersects the x - axis :

Options

A. 1
B. 2
C. 3
D. none of these

Answer

To determine how many times the graph of the function \( y = x^3 - 3x^2 - x + 3 \) intersects the x-axis, we need to find the roots of the equation \( y = 0 \). This means we need to solve the polynomial equation:

\[ x^3 - 3x^2 - x + 3 = 0 \]

### Step 1: Find a root of the polynomial
We can start by testing some simple values of \( x \) to see if they yield \( y = 0 \).

Let's try \( x = 1 \):

\[ y = 1^3 - 3(1^2) - 1 + 3 = 1 - 3 - 1 + 3 = 0 \]

Since \( y = 0 \) when \( x = 1 \), we have found one root: \( x = 1 \).

### Step 2: Factor the polynomial
Since \( x = 1 \) is a root, we can factor the polynomial using \( (x - 1) \). We will perform polynomial long division to divide \( x^3 - 3x^2 - x + 3 \) by \( (x - 1) \).

1. Divide \( x^3 \) by \( x \) to get \( x^2 \).
2. Multiply \( (x - 1) \) by \( x^2 \) to get \( x^3 - x^2 \).
3. Subtract \( (x^3 - x^2) \) from \( (x^3 - 3x^2) \) to get \( -2x^2 \).
4. Bring down the next term, resulting in \( -2x^2 - x + 3 \).
5. Divide \( -2x^2 \) by \( x \) to get \( -2x \).
6. Multiply \( (x - 1) \) by \( -2x \) to get \( -2x^2 + 2x \).
7. Subtract \( (-2x^2 + 2x) \) from \( (-2x^2 - x) \) to get \( -3x + 3 \).
8. Bring down the next term, resulting in \( -3x + 3 \).
9. Divide \( -3x \) by \( x \) to get \( -3 \).
10. Multiply \( (x - 1) \) by \( -3 \) to get \( -3x + 3 \).
11. Subtract to get \( 0 \).

Thus, we have:

\[ x^3 - 3x^2 - x + 3 = (x - 1)(x^2 - 2x - 3) \]

### Step 3: Factor the quadratic
Next, we need to factor the quadratic \( x^2 - 2x - 3 \). We can factor it as follows:

\[ x^2 - 2x - 3 = (x - 3)(x + 1) \]

### Step 4: Write the complete factorization
Now we can write the complete factorization of the polynomial:

\[ y = (x - 1)(x - 3)(x + 1) \]

### Step 5: Find the roots
To find the x-intercepts (roots), we set each factor equal to zero:

1. \( x - 1 = 0 \) gives \( x = 1 \)
2. \( x - 3 = 0 \) gives \( x = 3 \)
3. \( x + 1 = 0 \) gives \( x = -1 \)

### Conclusion
The roots of the polynomial are \( x = 1, x = 3, \) and \( x = -1 \). Therefore, the graph of the function intersects the x-axis **three times**.

### Final Answer
The graph of \( y = x^3 - 3x^2 - x + 3 \) intersects the x-axis **3 times**.

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