Problem 67 - A Graphical Approach to Precalculus with Limits

Question

For each polynomial at least one zero is given. Find all others analytically. $$P(x)=x^{4}-41 x^{2}+180 ; \quad-6 \text { and } 6$$

Step-by-Step Solution

Answer

The zeros are -6, 6, \(\sqrt{5}\), and \(-\sqrt{5}\).

Step 1: Understand the Given Polynomial and Known Zeros

We are given the polynomial \(P(x) = x^4 - 41x^2 + 180\) and two known zeros, \(x = -6\) and \(x = 6\). This means \((x + 6)\) and \((x - 6)\) are factors of \(P(x)\).

Step 2: Create a Factored Expression Using Known Zeros

Since \((x + 6)\) and \((x - 6)\) are factors, we can multiply them together to get \((x^2 - 36)\). Thus, \(P(x)\) can be expressed as \(P(x) = (x^2 - 36)Q(x)\) for some polynomial \(Q(x)\).

Step 3: Divide the Polynomial by the Product of Known Factors

Perform polynomial division of \(x^4 - 41x^2 + 180\) by \(x^2 - 36\) to determine \(Q(x)\). This division gives \(Q(x) = x^2 - 5\).

Step 4: Find Additional Zeros from New Quadratic Polynomial

Now find the zeros of \(Q(x) = x^2 - 5\). Set \(x^2 - 5 = 0\), so \(x^2 = 5\). Taking the square root of both sides, we find \(x = \pm\sqrt{5}\).

Step 5: Compile All Zeros of the Polynomial

The zeros of the polynomial \(P(x)\) are \(-6, 6\), and the roots of \(x^2 - 5\), which are \(\sqrt{5}\) and \(-\sqrt{5}\). So, the complete set of zeros is \(-6, 6, \sqrt{5}, -\sqrt{5}\).

Key Concepts

Polynomial Division

Polynomial division is akin to long division with numbers. It's used to divide one polynomial by another, helping to break complex expressions into simpler components. In our exercise, we began with the polynomial \( P(x) = x^{4} - 41x^{2} + 180 \). Given known zeros, -6 and 6, we established initial factors, \( (x + 6) \) and \( (x - 6) \), which when multiplied give \( x^{2} - 36 \).

To find other factors, we performed polynomial division, dividing \( x^{4} - 41x^{2} + 180 \) by \( x^{2} - 36 \). The result, \( Q(x) = x^{2} - 5 \), revealed more about our polynomial.

Here are steps to perform polynomial division:

• Write out the dividend and divisor.
• Divide the leading term of the dividend by the leading term of the divisor.
• Multiply the entire divisor by this quotient and subtract from the dividend.
• Repeat until reaching a remainder of zero or a smaller degree than the divisor.

This process not only simplifies but also aids in finding hidden factors or roots of the polynomial.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors, typically simpler binomials or trinomials. With the given polynomial, \( P(x) = x^{4} - 41x^{2} + 180 \), we knew two zeros that correlate to factors, \( (x + 6)(x - 6) = x^{2} - 36 \).

Thus, we could rewrite the polynomial as a product:

• First factor: \( (x^{2} - 36) \) from the known zeros.
• Second factor: \( (x^{2} - 5) \), obtained through polynomial division.

The factored form \( P(x) = (x^{2} - 36)(x^{2} - 5) \) simplifies solving for zeros since each quadratic can individually be set to zero. Factorization is a powerful technique because it decomposes complex polynomials, making them easier to analyze and solve.

Quadratic Equations

Quadratic equations are equations of the form \( ax^{2} + bx + c = 0 \). In our example, through division, we obtained \( x^{2} - 5 = 0 \) which is a simple quadratic equation.

To solve a quadratic equation, use methods like factoring, completing the square, or the quadratic formula. Here, we took:

• Directly set \( x^{2} - 5 = 0 \) and solve for \( x \).
• Isolate \( x^{2} = 5 \).
• Take square roots to find \( x = \pm\sqrt{5} \).

Simplifying quadratics is often straightforward, and zero finding becomes clear with each squared term handled separately.

Zero Finding

Zero finding is about identifying the roots of the polynomial—where it equals zero. Knowing zeros allows us to factor and break down polynomials effectively. Given and discovered zeros of \( P(x) \) include -6, 6, and \( \pm\sqrt{5} \).

To find zeros, start by recognizing existing or given zeros. These correspond to clear factors of the polynomial, informing our strategy of division and simplification. Once the equation is reduced through division:

• Set each factor equals zero.
• Solve for the variable \( x \).
• Collect all solutions for a complete root set.

Zero finding is crucial as it confirms the roots and ensures the polynomial's integrity when factored back. Recognizing all zeros, in combination with polynomial division or factoring, bridges complex problems to clear solutions.