Problem 68 - A Graphical Approach to Precalculus with Limits

Question

For each polynomial at least one zero is given. Find all others analytically. $$P(x)=x^{4}-52 x^{2}+147 ; \quad-7 \text { and } 7$$

Step-by-Step Solution

Answer

The zeros of the polynomial are \(-7, 7, \sqrt{3},\) and \(-\sqrt{3}\).

Step 1: Understanding the Problem

We are given a polynomial \( P(x) = x^4 - 52x^2 + 147 \) with known zeros \(-7\) and \(7\). These zeros indicate that \((x + 7)\) and \((x - 7)\) are factors of \(P(x)\). Our task is to find all the zeros of the polynomial.

Step 2: Express the Polynomial Using Known Zeros

Since \(-7\) and \(7\) are zeros of \(P(x)\), \( (x + 7)(x - 7) = x^2 - 49 \) is a factor of \(P(x)\). Therefore, \( P(x) \) can be expressed as \( (x^2 - 49)(x^2 + bx + c) \). We will find the values of \(b\) and \(c\) by polynomial division.

Step 3: Polynomial Division

Divide \( P(x) = x^4 - 52x^2 + 147 \) by \( x^2 - 49 \) using polynomial long division. The quotient will be a quadratic polynomial \(x^2 + bx + c\).

Step 4: Set Up and Perform the Division

1. The first term in the quotient will be \(x^2\) because \(x^2 \cdot (x^2 - 49) = x^4 - 49x^2\).2. Subtract \(x^4 - 49x^2\) from \(P(x)\) to get \(-3x^2 + 147\).3. The next term in the quotient is \(-3\) because \(-3 \cdot (x^2 - 49) = -3x^2 + 147\).4. Subtract \(-3x^2 + 147\) from \(-3x^2 + 147\) to get 0.So \(P(x) = (x^2 - 49)(x^2 - 3)\).

Step 5: Solve for the Remaining Zeros

Now, solve \(x^2 - 3 = 0\). This simplifies to \(x^2 = 3\), so the solutions are \(x = \sqrt{3}\) and \(x = -\sqrt{3}\).

Step 6: List All Zeros

The zeros of the polynomial \(P(x) = x^4 - 52x^2 + 147\) are \(-7, 7, \sqrt{3},\) and \(-\sqrt{3}\).

Key Concepts

Polynomial Division

Polynomial division is a technique that allows us to divide one polynomial by another, similar to how you divide numbers. In this process, you reduce a larger polynomial into simpler components, which helps in finding factors and zeros.
When given a polynomial like \(P(x) = x^4 - 52x^2 + 147\), with some known zeros, polynomial division is crucial to simplify the polynomial into smaller parts.

• Identify the divisor, which in this case, is \(x^2 - 49\).
• The goal is to divide \(P(x)\) by this divisor, reducing it to a simpler polynomial.

Through long division, subtract terms in sequence, allowing each step to bring you closer to the simplified polynomial form. This helps spot any new factors that provide more zeros, as shown when dividing \(x^4 - 52x^2 + 147\) by \(x^2 - 49\), resulting in a quotient of \(x^2 - 3\).
This technique is very powerful in solving polynomial equations, making it easier to factorize complex polynomials.

Quadratic Factors

Quadratic factors are expressions of the form \(ax^2 + bx + c\). They are an essential part of understanding and solving polynomial equations, especially when the polynomial is of degree four or higher.
In our exercise, once we simplified the original polynomial with division, we found a quadratic equation, \(x^2 - 3\). This indicates new potential zeros for \(P(x)\).

• Identifying these quadratic factors lets us break down polynomials further into their roots.
• Every quadratic factor can potentially give two zeros, simplifying the task of finding all solutions.

By solving \(x^2 - 49 = 0\) and \(x^2 - 3 = 0\), we get the zeros for these factors separately. \(x^2 - 49\) resolves to \(-7\) and \(7\), while \(x^2 - 3\) leads to solutions \(\sqrt{3}\) and \(-\sqrt{3}\). These factors make finding zeros a straightforward task, turning complex polynomials into manageable equations.

Factoring

Factoring allows us to express a polynomial as a product of simpler polynomials. This process is crucial when solving equations as it reveals the zeros of the polynomial.
When you factor \(P(x) = x^4 - 52x^2 + 147\), you start by knowing the zeros \(-7\) and \(7\) which helps you determine factors \((x+7)\) and \((x-7)\).

• Once the polynomial is reduced using division, each factor corresponds to a potential zero.
• The core steps include checking all components, rearranging them into a product, and verifying the correctness.

After reducing \(P(x)\) using \(x^2 - 49\), you find it as \((x+7)(x-7)(x^2 - 3)\). Solving each factor separately for zero gives complete solutions: \(-7, 7, \sqrt{3}, -\sqrt{3}\). Successful factoring leads to a set of zeros that solve the polynomial equation. It simplifies complex expressions and is often the most direct route to uncovering all possible polynomial zeros.