Problem 52 - College Algebra and Calculus: An Applied Approach

Question

Use the given zero of \(f\) to find all the zeros of \(f\). $$f(x)=x^{3}-7 x^{2}-x+87, \quad 5+2 i$$

Step-by-Step Solution

Answer

The roots of the function \(f(x) = x^{3}-7 x^{2}-x+87\) are \(5+2i\), \(5-2i\) and the roots of the resulting quadratic equation found in Step 3.

Step 1: Identify the complex conjugate

The polynomial has real coefficients. Therefore, if \(5+2i\) is a root, then its complex conjugate \(5-2i\) is also a root. Complex conjugates appear as roots in pairs when the polynomial has real coefficients. So, besides \(5+2i\), we also have \(5-2i\) as a second root of the polynomial.

Step 2: Perform polynomial long division

Now we divide the polynomial \(f(x) = x^{3}-7 x^{2}-x+87\) by the quadratic expression generated by the roots \(5+2i\) and \(5-2i\). This expression is \((x-(5+2i)) \cdot (x-(5-2i)) = (x-5-2i)(x-5+2i) = (x-5)^2 - (2i)^2 = x^2 - 10x + 29\). The result of this long division is the remaining polynomial which will already have only real roots.

Step 3: Solve the resulting quadratic equation

Solving the polynomial resulting from the long division \(x^2 - 10x + 29 = 0\), we use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) to find the last pair of any complex or real roots.

Key Concepts

Complex Conjugate Roots

When dealing with polynomials, a common occurrence is finding complex roots, especially when the polynomial has real coefficients. According to the Complex Conjugate Root Theorem, if a polynomial has real coefficients and a complex number like \( a + bi \) is a root, its complex conjugate \( a - bi \) must also be a root. This is a fundamental concept since it allows us to identify a pair of roots quickly.For instance, in our exercise, we are given that \( 5+2i \) is a root of \( f(x)=x^{3}-7x^{2}-x+87 \). Given that the polynomial’s coefficients are real, we can immediately deduce that the complex conjugate root, \( 5-2i \), is also a root.

• Complex conjugate roots come in pairs - if you find one, you know the other immediately.
• This theorem only applies when all the coefficients of the polynomial are real numbers.
• Knowing both of the complex conjugate roots can help when simplifying or factoring the polynomial.

Understanding this concept is crucial to finding all zeros of a polynomial, as it simplifies the process and reduces the amount of calculation needed.

Polynomial Long Division

Polynomial long division is a technique similar to long division with numbers, used to divide a polynomial by another polynomial of lesser or equal degree. The goal is to find the quotient and the remainder. This method is particularly useful when trying to factor a polynomial after knowing one of its roots or when simplifying a rational expression.

Applying Polynomial Long Division

In our given problem, after knowing that \( 5+2i \) and \( 5-2i \) are roots, we use their conjugates to construct a quadratic \( (x-(5+2i))(x-(5-2i)) \), which simplifies to \( x^2 - 10x + 29 \). We then perform long division by dividing the original polynomial \( f(x) \) by this quadratic factor.

• The result of this division gives us a reduced polynomial which is easier to solve.
• It often leads to a quadratic polynomial if you're starting with a cubic one by dividing out the known roots.
• If correctly done, the division should leave no remainder if the roots used are indeed correct.

Recognizing when and how to apply polynomial long division is essential for breaking down complex polynomials into more manageable pieces.

Quadratic Formula

One of the most reliable tools for finding the zeros of a quadratic equation is the quadratic formula. The formula states that for any quadratic equation in the form of \( ax^2 + bx + c = 0 \), the solutions can be found using:\[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \].

Applying the Quadratic Formula

For our exercise, once we have divided out the complex roots and are left with a quadratic equation \( x^2 - 10x + 29 = 0 \), we use the quadratic formula to find the remaining zeros of the polynomial. By plugging the coefficients \( a = 1 \), \( b = -10 \), and \( c = 29 \) into the formula, we can solve for \( x \).

• This formula works for all quadratic equations, whether or not the roots are real or complex.
• The term under the square root, \( b^2-4ac \), is called the discriminant and determines the nature of the roots.
• A positive discriminant indicates two real and distinct roots; zero means one repeated real root, and negative signifies two complex roots.

The quadratic formula is a powerful tool for students to quickly and accurately find the roots of quadratic equations without resorting to completing the square or factoring.