Problem 39 - College Algebra

Question

In Exercises \(37-44,\) find all the zeros of the function and write the polynomial as a product of linear factors. $$ f(x)=x^{3}-8 x^{2}+25 x-26 $$

Step-by-Step Solution

Answer

The zeros of the function are \(x = 1, 3.5 + 0.5i, 3.5 - 0.5i\). The polynomial as a product of linear factors is \(f(x) = (x - 1)(x^2 - 7x + 26)\).

Step 1: Solve the Equation

The first step involves setting up the equation to find the zeros of the function. So, set \(f(x) = 0\): \(x^{3}-8 x^{2}+25 x-26 = 0\) Now, the task is to solve this equation for \(x\).

Step 2: Factorizing the Polynomial

The next step involves finding the correct factors of the polynomial so that we can identify the roots easily. Begin by trialling various values to find a root. For example, \(x = 1\) will work, since \(1^3 - 8 * 1^2 + 25 * 1 - 26 = 0\). Therefore, \(1\) is a root of the polynomial. Perform polynomial division (or use the factor theorem) to confirm this and find the quadratic factor. This yields: \((x - 1)(x^2 - 7x + 26) = 0\)

Step 3: Finding More Roots

Now, after finding the first root, we need to identify other roots. To do that, we set the quadratic factor equal to zero and solve for \(x\): \(x^2 - 7x + 26 = 0\). Use the quadratic formula to find \(x = (7 \pm \sqrt{7^2 - 4*26})/2 = (7 \pm \sqrt{-1})/2\), which means two of the zeros are complex numbers: \(x = 3.5 \pm 0.5i\).

Step 4: Representing as Linear Factors

Finally, express the given polynomial as a product of linear factors. Because two of the zeros are complex conjugates, \(x-1\) accounts for one of the linear factors and \((x^2 - 7x + 26)\) accounts for the other two. Therefore, our original third degree polynomial is represented as follows: \(f(x) = (x - 1)(x^2 - 7x + 26)\)

Key Concepts

Zeros of a Function

The zeros of a function are the values of the variable that make the function equal to zero. In other words, if you substitute a zero into the function, it results in zero. Finding the zeros of a polynomial function like \(f(x) = x^3 - 8x^2 + 25x - 26\) means solving the equation \(f(x) = 0\).

• These zeros represent the x-intercepts of the polynomial graph on the coordinate plane.
• For our exercise, we found that one zero is \(x = 1\) by checking the equation with different values.
• The other zeros for our polynomial are found using the quadratic formula, as the remaining part of the polynomial after factorizing \((x-1)\) requires solving the quadratic equation \(x^2 - 7x + 26 = 0\).

Understanding zeros helps us to factorize the polynomial effectively and graph its behavior.

Complex Numbers

Complex numbers often arise when dealing with polynomial equations that do not cross the x-axis or when the discriminant in a quadratic formula is negative. A standard complex number looks like this: \(a + bi\).

• Here, \(a\) is the real part, and \(bi\) is the imaginary part.
• In our polynomial factorization problem, the quadratic equation \(x^2 - 7x + 26 = 0\) yields complex numbers as its solutions due to a negative discriminant (a value under the square root in the quadratic formula that is negative).
• The solutions are \(x = 3.5 \pm 0.5i\), where these numbers don't cross the x-axis but are still crucial for fully factorizing the polynomial.
• These complex zeros appear as conjugate pairs, ensuring the polynomial remains real-valued as required for a function with real coefficients.

Managing complex numbers allows for the complete factorization and understanding of higher degree polynomials.

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of any quadratic equation in the form \(ax^2 + bx + c = 0\). The formula is given as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This equation is derived from completing the square and helps in determining whether the roots are real, repeated, or complex.

• The term \(b^2 - 4ac\) is called the discriminant.
• If the discriminant is positive, there are two real roots; if zero, there's one repeated real root; and if negative, the roots are complex.
• In our polynomial factorization problem, the quadratic \(x^2 - 7x + 26\) had a negative discriminant, hinting at complex roots.
• By substituting values into the quadratic formula, we found the two complex roots: \(x = 3.5 \pm 0.5i\).

Utilizing the quadratic formula is essential for solving and understanding the behaviors of quadratic components within a polynomial.