Problem 67 - A Graphical Approach to College Algebra

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 of the polynomial are \(-6, 6, \sqrt{5}, -\sqrt{5}\).

Step 1: Identify the Given Zeros

You are given that the polynomial function is \(P(x) = x^4 - 41x^2 + 180\) and that \(x = -6\) and \(x = 6\) are zeros of the polynomial. This means \(P(-6) = 0\) and \(P(6) = 0\). Since the polynomial is an even function (the powers of \(x\) are even), it is symmetric. The polynomial must also be divisible by both \((x + 6)\) and \((x - 6)\).

Step 2: Use Factoring Approach

Since \(x = -6\) and \(x = 6\) are zeros, a factor of \(P(x)\) is \((x^2 - 36)\), because \((x - 6)(x + 6) = x^2 - 36\). Divide the polynomial by this factor to find the remaining factors. Perform the division:\[\frac{x^4 - 41x^2 + 180}{x^2 - 36}\].

Step 3: Perform Polynomial Division

Divide \(x^4 - 41x^2 + 180\) by \(x^2 - 36\) using polynomial long division. Start by dividing \(x^4\) by \(x^2\) to get \(x^2\). Multiply \(x^2\) with \(x^2 - 36\) to get \(x^4 - 36x^2\). Subtract from the original polynomial to get \(-5x^2 + 180\).

Step 4: Complete the Division

Continue the division by dividing \(-5x^2\) by \(x^2\) to get \(-5\). Multiply \(-5\) by \(x^2 - 36\) to get \(-5x^2 + 180\). Subtract this from \(-5x^2 + 180\) to get a remainder of 0, indicating \(P(x) = (x^2 - 36)(x^2 - 5)\).

Step 5: Solve for Remaining Zeros

Now solve \(x^2 - 5 = 0\) to find the remaining zeros. Rewrite as \(x^2 = 5\), then take the square root of both sides to get \(x = \sqrt{5}\) and \(x = -\sqrt{5}\). Therefore, the zeros of the polynomial are \(x = -6, 6, \sqrt{5}, -\sqrt{5}\).

Key Concepts

Polynomial Division

Polynomial Division is a technique used to divide one polynomial by another, often resulting in a quotient and a remainder. In this exercise, we are dividing the polynomial \(P(x) = x^4 - 41x^2 + 180\) by the factor \(x^2 - 36\). Start by aligning the terms of \(x^4 - 41x^2 + 180\) with \(x^2 - 36\).

• First, divide the leading term \(x^4\) by \(x^2\), resulting in \(x^2\).
• Next, multiply \(x^2\) by the divisor \(x^2 - 36\), which gives \(x^4 - 36x^2\).
• Subtract this from the original polynomial to simplify.
• This process continues until the entire polynomial is divided, ensuring the remainder is zero when perfectly divisible.

Polynomial division is crucial as it helps identify factors of polynomials and finds roots, thereby simplifying complex expressions and solving polynomial equations.

Factoring Polynomials

Factoring Polynomials is the process of breaking down a polynomial into simpler components or factors that, when multiplied together, rapidly form the original polynomial.In the exercise, once we identified the zeros, \(x = -6\) and \(x = 6\), we found a common factor: \((x + 6)(x - 6) = x^2 - 36\). This illustrates the symmetry in polynomials, particularly when dealing with even functions. After discovering \(x^2 - 36\) as a factor, we performed polynomial division on \(P(x)\) to further factor it as \((x^2 - 36)(x^2 - 5)\).

• Recognizing patterns, like squares and differences of squares, can simplify factorizations.
• Using given zeros to create factors often leads to uncovering additional zeros.

Factoring is a powerful tool in mathematics, beneficial for solving equations and simplifying expressions.

Roots of Polynomials

Roots or zeros of a polynomial are the values of \(x\) that make the polynomial equal to zero. In essence, they are where the polynomial crosses the x-axis. Finding these roots involves several techniques, including substitution, factorizing, and using the quadratic formula.For the given polynomial \(P(x) = x^4 - 41x^2 + 180\), we were initially provided with two zeros: \(x = -6\) and \(x = 6\). By factoring, we found additional zeros from \(x^2 - 5 = 0\), calculated as \(x = \sqrt{5}\) and \(x = -\sqrt{5}\).

• Zeros are critical as they reveal the solutions to the polynomial equation.
• They help in sketching the graph of the polynomial, demonstrating its behavior across different intervals.
• Understanding the relationship between roots and factors expedites the solving of complex polynomials.

By comprehending these concepts, students can effectively find polynomial solutions and apply these insights to real-world mathematical problems.