Problem 79 - A Graphical Approach to Precalculus with Limits

Question

The solutions to the equation \(x^{3}-85 x+300=0\) were found graphically. These solutions can be found exactly by using analytic methods, as shown in the next two exercises. Use synthetic division to show that 5 is a zero of \(x^{3}-85 x+300 .\) Rewrite this polynomial by factoring out \((x-5)\)

Step-by-Step Solution

Answer

The polynomial \(x^3 - 85x + 300\) can be factored as \((x - 5)(x^2 + 5x - 60)\).

Step 1: Set Up Synthetic Division

To perform synthetic division, first set up the synthetic division table. For the polynomial \(x^3 - 85x + 300\), we only have the terms involving \(x^3\), \(x\), and the constant term. Write the coefficients for \(x^3\), \(x\), and the constant as: \([1, 0, -85, 300]\). Since we need to check that \(5\) is a zero, place \(5\) to the left of the table.

Step 2: Perform Synthetic Division

Begin the synthetic division process:1. Bring down the first coefficient directly below the line: \(1\).2. Multiply this number by \(5\) and write the result under the next coefficient (0), perform the addition: \(1 \times 5 = 5\), place \(5\) under \(0\).3. Add the values: \(0 + 5 = 5\).4. Repeat the process: Multiply \(5\) by \(5\): \(5 \times 5 = 25\), place this under \(-85\).5. Add the values: \(-85 + 25 = -60\).6. Repeat: Multiply \(-60\) by \(5\): \(-60 \times 5 = -300\), place this under \(300\).7. Add the values: \(300 - 300 = 0\).The remainder is \(0\), confirming \(5\) is a zero of the polynomial.

Step 3: Interpret the Results

Since the remainder is \(0\), divide the original polynomial by \(x - 5\) successfully. The synthetic division yields the quotient polynomial \(x^2 + 5x - 60\). Write the original polynomial as \((x - 5)(x^2 + 5x - 60)\).

Key Concepts

Polynomial Division

Polynomial Division is a method used to divide polynomials, much like long division with numbers. It helps break down complex polynomials into simpler parts. This method can be performed using either long division or synthetic division.

• In Polynomial Division, the dividend is the polynomial we are dividing, and the divisor is what we divide by.
• The result of the division is called the quotient, and sometimes there is a remainder.

One common technique is Synthetic Division, which is sometimes preferred for its simplicity and speed. It is especially useful when dividing by a linear binomial, such as \(x - c\). This method only uses the coefficients of the polynomial, making it less complex than long division in many cases.
By knowing how to perform synthetic division, you can easily test if a number is a zero of a polynomial and also factor the polynomial efficiently.

Factorization

Factorization involves breaking down a polynomial into a product of its factors. It is similar to expressing a number as a product of its prime factors. When working with polynomials, factorization helps to understand the structure of the polynomial and efficiently solve equations.

• A factor of a polynomial is a polynomial that divides the original polynomial without leaving a remainder.
• Factoring polynomials can help simplify expressions and solve polynomial equations more easily.

In the example provided, once synthetic division confirms that 5 is a zero, the polynomial \(x^3 - 85x + 300\) can be factored as \((x - 5)(x^2 + 5x - 60)\).
This step is crucial as it reduces the complexity of the polynomial, making further analysis and finding more factors or zeros simpler.

Zeros of Polynomials

The zeros of a polynomial are the values of \(x\) that make the polynomial equal to zero. Finding these values is fundamental in solving polynomial equations.

• Zeros can also be referred to as roots or solutions of the polynomial equation.
• For a polynomial of degree \(n\), there can be up to \(n\) zeros.

To find the zeros, you can use several techniques such as:

• Factoring the polynomial and solving the equation using its factors
• Performing polynomial division, like synthetic division, to find potential zeros
• Using the quadratic formula for second-degree polynomials

In the exercise, synthetic division shows 5 as a zero of \(x^3 - 85x + 300\). Once zero is identified, the polynomial can be factored further to find additional zeros, enhancing the understanding and ability to solve the equation.