Problem 14 - A Graphical Approach to College Algebra

Question

Find a polynomial function \(P(x)\) of degree 3 with real coefficients that satisfies the given conditions. Do not use a calculator. \(P(x)=x^{3}-5 x^{2}+17 x-13 ; \quad 1\) is a zero.

Step-by-Step Solution

Answer

The polynomial is already given and meets the conditions; it is \(P(x) = (x - 1)(x^2 - 4x + 13)\).

Step 1: Use the Given Zero

We are given that 1 is a zero of the polynomial function \(P(x) = x^3 - 5x^2 + 17x - 13\). This means that \(P(1) = 0\). Check this by substituting 1 into the polynomial.

Step 2: Perform Synthetic Division

Since 1 is a zero, \(x - 1\) is a factor of \(P(x)\). Use synthetic division to divide the polynomial by \(x - 1\). List the coefficients: 1 (for \(x^3\)), -5 (for \(x^2\)), 17 (for \(x\)), and -13 (constant term). Perform the synthetic division.

Step 3: Analyze the Result

After performing synthetic division, you find that the division is exact with a remainder of 0, confirming \(x - 1\) is a factor. The quotient gives us a quadratic polynomial.

Step 4: Factor the Quadratic

The result from synthetic division is a quadratic polynomial, say \(Q(x)\). Factor \(Q(x)\) if possible, to find the remaining factors of \(P(x)\).

Step 5: Write the Final Polynomial Function

Compile the factors from step 2 and 4 to write the polynomial function in factor form. Each factor corresponds to a zero of the polynomial.

Key Concepts

Synthetic Division

Synthetic division is a simplified method to divide a polynomial by a binomial of the form \( x - c \). Compared to long division, this method is not only quicker but also less prone to mistakes. Here's how we tackle it in three simple steps:

• Identify the Zero: Start by identifying the zero, \( c \). In our example, since 1 is given as zero, \( c = 1 \).
• Write the Coefficients: List down the coefficients of the polynomial you're dividing. For \( P(x) = x^3 - 5x^2 + 17x - 13 \), the coefficients are 1, -5, 17, and -13.
• Perform the Division: Using these coefficients, perform synthetic division by setting the zero in a division box and applying straightforward arithmetic to get the remainder.

If the remainder is zero, it confirms that \( x - c \) is a factor of the polynomial. This is a crucial result that simplifies finding further factors or solving polynomial equations.

Zero of a Polynomial

The zero of a polynomial, sometimes called a root, is a value that makes the polynomial equal to zero. In our case, given that \( P(1) = 0 \), 1 is a confirmed zero.
Understanding zeros is fundamental because:

• Each zero indicates where the polynomial will intersect the x-axis on a graph.
• Finding one zero can assist in calculating others, due to the polynomial's factorability using synthetic division or other methods.

When you know that \( x - c \) is a factor thanks to a zero like 1, the division of the polynomial by \( x - c \) (using synthetic division) provides a simpler polynomial of a lower degree, further aiding in finding additional zeros or factors of the original polynomial.

Factorization

Factorization involves breaking down a polynomial into a product of its factors, which often includes linear or quadratic factors. From the earlier synthetic division of \( P(x) = x^3 - 5x^2 + 17x - 13 \), we confirm that \( x - 1 \) is a factor since it divides the polynomial without leaving a remainder.
Once synthetic division gives a quotient polynomial:\[ Q(x) = ax^2 + bx + c \]
You would next attempt to factor this quadratic polynomial if possible. Depending on its complexity, it might involve:

• Completing the square,
• Using the quadratic formula,
• Simple inspection for factors when coefficients are small.

Factorization reveals the other zeros of the polynomial and helps express it fully in its factorized form, making it advantageous for solving, graphing, or further algebraic manipulation.