Verify that $3, -1, -\dfrac{1}{3}$ are the zeroes of the cubic polynomial $p(x) = 3x^3 - 5x^2 - 11x - 3$, and then verify the relationship between the zeroes and the coefficients.

Step 1: Evaluate $p(x)$ at $x=3$

To verify that a number is a zero of a polynomial, we substitute that number into the polynomial and check if the result is zero. We will start by substituting $x=3$ into the polynomial $p(x) = 3x^3 - 5x^2 - 11x - 3$.

Step 2: Calculate $p(3)$

After substituting $x=3$, we perform the arithmetic operations: $3^3 = 27$, $3^2 = 9$. Then, we calculate the products and sums to find the value of $p(3)$. Since $p(3)=0$, $x=3$ is indeed a zero of the polynomial.

Step 3: Evaluate $p(x)$ at $x=-1$

Next, we substitute $x=-1$ into the polynomial $p(x)$ to check if it is also a zero. We replace every $x$ in the expression with $-1$.

Step 4: Calculate $p(-1)$

We calculate $(-1)^3 = -1$ and $(-1)^2 = 1$. Then we perform the multiplications and additions. Since $p(-1)=0$, $x=-1$ is also a zero of the polynomial.

Step 5: Evaluate $p(x)$ at $x=-\frac{1}{3}$

Finally, we substitute $x=-\frac{1}{3}$ into the polynomial $p(x)$ to verify the last given zero. We carefully substitute the fractional value into the expression.

Step 6: Calculate $p(-\frac{1}{3})$

We calculate $(-\frac{1}{3})^3 = -\frac{1}{27}$ and $(-\frac{1}{3})^2 = \frac{1}{9}$. Then we simplify the expression by finding a common denominator for the fractions. Since $p(-\frac{1}{3})=0$, $x=-\frac{1}{3}$ is also a zero of the polynomial. Thus, all three given values are verified zeros.

Step 7: Identify coefficients of $p(x)$

Now we will verify the relationship between the zeroes and the coefficients. A general cubic polynomial is given by $ax^3 + bx^2 + cx + d$. By comparing this with $p(x) = 3x^3 - 5x^2 - 11x - 3$, we can identify the coefficients $a, b, c, d$.

Step 8: Assign values to zeroes

Let the zeroes be $\alpha$, $\beta$, and $\gamma$. We assign the given verified zeroes to these variables for convenience in applying the relationships.

Step 9: Verify sum of zeroes

The relationship for the sum of the zeroes of a cubic polynomial is $\alpha + \beta + \gamma = -\frac{b}{a}$. We calculate the sum of our zeroes and compare it with $-b/a$ using the coefficients identified earlier. Both values match, verifying this relationship.

Step 10: Verify sum of products of zeroes taken two at a time

The relationship for the sum of the products of the zeroes taken two at a time is $\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}$. We compute this sum using our zeroes and compare it with $c/a$. Both results are equal, confirming this relationship.

Step 11: Verify product of zeroes

The relationship for the product of the zeroes of a cubic polynomial is $\alpha\beta\gamma = -\frac{d}{a}$. We calculate the product of our zeroes and compare it with $-d/a$. The values match, which verifies the final relationship.

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