Problem 18 - A Graphical Approach to College Algebra

Question

For each polynomial, one or more zeros are given. Find all remaining zeros. \(P(x)=2 x^{4}-2 x^{3}+55 x^{2}-50 x+125 ; \quad-5 i\) is a zero.

Step-by-Step Solution

Answer

Remaining zeros are complex conjugates -5i, 5i. Solve quadratic for real zeros.

Step 1: Identify conjugate pair of zeros

Since the polynomial has real coefficients, any non-real complex zeros must occur in conjugate pairs. Given that \(-5i\) is a zero, its conjugate \(5i\) must also be a zero.

Step 2: Form polynomial from zeros

Use the zeros \(-5i\) and \(5i\) to form a quadratic polynomial:\( (x + 5i)(x - 5i) = x^2 + 25 \). This is derived using the difference of squares formula.

Step 3: Polynomial division

Divide the original polynomial \(P(x) = 2x^4 - 2x^3 + 55x^2 - 50x + 125\) by the quadratic \(x^2 + 25\). Perform synthetic or long division to find the quotient polynomial.

Step 4: Find remaining zeros

The resultant polynomial from the division will be a quadratic. Solve this quadratic using the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \], where \(a\), \(b\), and \(c\) are coefficients of the quadratic. This will give the remaining zeros.

Key Concepts

Complex Conjugate Zeros

In the realm of polynomials, zeros are the values of \(x\) for which the polynomial evaluates to zero. When dealing with polynomials that have real coefficients, one important property is that any non-real zeros must come in complex conjugate pairs. This means if \(-5i\) is a zero of a polynomial with real coefficients, its complex conjugate, \(5i\), must also be a zero. To further clarify this concept:

• A complex number is generally in the form \(a + bi\).
• Its conjugate is \(a - bi\).
• In this exercise, \(-5i\) corresponds to \(0 - 5i\), hence its conjugate is \(0 + 5i\), or \(5i\).

Understanding this property is crucial because it helps in reducing the degree of the polynomial by identifying two zeros at once whenever you find a non-real zero with a real-coefficient polynomial. This information is instrumental in simplifying the polynomial and finding all of its roots efficiently.

Quadratic Formula

The quadratic formula is a valuable tool for solving quadratic equations, which are polynomials of degree 2. It provides the roots of any quadratic equation of the form \(ax^2 + bx + c = 0\). The quadratic formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Using this formula allows you to calculate the zeros directly by substituting the coefficients \(a\), \(b\), and \(c\).
The discriminant in the formula, \(b^2 - 4ac\), determines the nature of the roots:

• Positive Discriminant: Results in two distinct real roots.
• Zero Discriminant: Results in one real root (double root).
• Negative Discriminant: Results in two complex conjugate roots.

In our exercise, after removing the known complex conjugate zeros, the remaining polynomial can be reduced to a quadratic.
This is where the quadratic formula comes into play to find any remaining zeros. By understanding this powerful equation, you can solve any quadratic efficiently, whether the roots are real or complex.

Polynomial Division

Polynomial division is a method used to simplify polynomials by dividing them. When finding the remaining zeros of a polynomial, we often need to divide by a known factor.For example, in our exercise, we already know that \(x^2 + 25\) is a factor due to the conjugate zeros \(-5i\) and \(5i\). Therefore, \(x^2 + 25\) divides the original polynomial \(P(x)\).There are generally two methods for polynomial division:

• Synthetic Division: A shortcut method easier than long division but only applicable when dividing by linear factors.
• Long Division: A more general technique similar to numerical long division, applicable for dividing by polynomials of any degree.

Once you perform the division, the original polynomial \(P(x)\) is reduced, and the quotient polynomial emerges. This quotient, often a quadratic, can then be further analyzed to find the remaining zeros of the original polynomial. Polynomial division thus transforms a potentially complex problem into more manageable steps, especially when paired with other tools like the quadratic formula.