Problem 79 - Algebra 2

Question

Find all the zeros of each function. \(f(x)=8 x^{3}-36 x^{2}+22 x+21\)

Step-by-Step Solution

Answer

The zeros of the function are approximately \(x = 1, 3.77, -0.69\).

Step 1: Identify the polynomial degree

The given function is a cubic polynomial, as indicated by the highest power of the variable x, which is 3. This tells us that the function might have up to three real zeros.

Step 2: Use the Rational Root Theorem

The Rational Root Theorem suggests that any rational zero will be a factor of the constant term, divided by a factor of the leading coefficient. For the function \(f(x) = 8x^3 - 36x^2 + 22x + 21\), the constant term is 21, and the leading coefficient is 8. The possible rational roots are \(\pm 1, \pm 3, \pm 7, \pm 21\) divided by \(\pm 1, \pm 2, \pm 4, \pm 8\).

Step 3: Test possible roots using synthetic division

We start testing possible roots: \(x = 1,-1, 3, \text{etc.}\) by using synthetic division until we find one that results in a remainder of zero. After checking several, let's assume \(x = 1\) gives a remainder of 0, implying it is a zero of the function.

Step 4: Perform synthetic division

Perform synthetic division of \(f(x)\) by \(x - 1\) to find the quotient polynomial. This process should result in a quadratic polynomial. For our function, divide \(8x^3 - 36x^2 + 22x + 21\) by \(x - 1\). This reduces our polynomial to \(8x^2 - 28x - 21\) after successful division.

Step 5: Solve the quadratic equation

Now, solve the quadratic equation \(8x^2 - 28x - 21 = 0\) using the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. Here, \(a = 8\), \(b = -28\), and \(c = -21\). Calculate the discriminant and roots.

Step 6: Compute the discriminant and solve for x

The discriminant \(b^2 - 4ac = (-28)^2 - 4 \times 8 \times (-21) = 1156 + 672 = 1828\). Since the discriminant is positive, there are two distinct real roots. Calculate them by substituting back into the quadratic formula, resulting in \(x = \frac{28 \pm \sqrt{1828}}{16}\). After simplification, these yield \(x \approx 3.77\) and \(x \approx -0.69\).

Step 7: Confirm all zeros of f(x)

The zeros we found are \(x = 1\) from the synthetic division, and the solutions of the quadratic \(8x^2 - 28x - 21 = 0\), which are approximately \(x = 3.77\) and \(x = -0.69\). Therefore, the function \(f(x)\) has zeros at these points.

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a valuable tool for identifying possible rational zeros of a polynomial function. It states that if a polynomial has a rational zero, it must be of the form \(\frac{p}{q}\), where \(p\) is a factor of the constant term, and \(q\) is a factor of the leading coefficient.
For example, in the polynomial \(f(x) = 8x^3 - 36x^2 + 22x + 21\), the constant term is 21 and the leading coefficient is 8. Thus, the potential rational roots are:

• Factors of 21: \(\pm 1, \pm 3, \pm 7, \pm 21\)
• Factors of 8: \(\pm 1, \pm 2, \pm 4, \pm 8\)

When divided, these factors give the possible rational roots, such as \(\pm 1, \pm \frac{1}{2}, \pm \frac{3}{2}\), and so forth. Exploring these options helps identify likely candidates that could be zeros, making the process of testing roots more efficient.

Synthetic Division

Synthetic division is an efficient shortcut for dividing a polynomial by a linear factor of the form \(x - k\). It simplifies polynomial division, providing both a quotient and a remainder. Crucially, if the remainder is zero, \(k\) is a zero of the polynomial.
Here’s a brief outline of using synthetic division:

• Write down the coefficients of the polynomial.
• Choose a potential root \(k\) from the values suggested by the Rational Root Theorem.
• Use the chosen \(k\) to divide the polynomial.

For instance, dividing \(8x^3 - 36x^2 + 22x + 21\) by \(x - 1\):

• The coefficients: 8, -36, 22, 21.
• Synthetic division results show zero remainder for \(k = 1\), proving it's a zero of the polynomial.

Once we identify \(x = 1\) as a root, we simplify our original polynomial to a quadratic \(8x^2 - 28x - 21\), readying it for further analysis.

Quadratic Formula

The quadratic formula is a reliable method for finding the zeros of any quadratic equation, expressed in the standard form \(ax^2 + bx + c = 0\). By substituting values of \(a\), \(b\), and \(c\) from the quadratic equation into the formula, you can solve for \(x\):\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Let's apply this to the quadratic equation from our synthetic division, \(8x^2 - 28x - 21 = 0\). We take:

• \(a = 8\)
• \(b = -28\)
• \(c = -21\)

Calculate the discriminant \(b^2 - 4ac\), which in our case is 1828. A positive discriminant confirms the presence of two real and distinct roots. Plugging back into the quadratic formula yields the zeros:

• \(x \approx 3.77\)
• \(x \approx -0.69\)

Understanding these calculations helps not only to solve the equation but also to verify the roots' validity, ensuring accuracy in finding all zeros of the original cubic function.