Problem 25 - College Algebra and Calculus: An Applied Approach

Question

Find all the zeros of the function and write the polynomial as a product of linear factors. $$f(x)=16 x^{3}-20 x^{2}-4 x+15$$

Step-by-Step Solution

Answer

The zeros of the function \(f(x) = 16x^3 - 20x^2 -4x +15\) are \(x = -1, -\frac{3}{4}, \frac{5}{4}\) and the polynomial can be expressed as \(f(x) = 16(x+1)(x+\frac{3}{4})(x-\frac{5}{4})\).

Step 1: Identify Potential Rational Roots Using the Rational Root Theorem

The Rational Root theorem states that any possible rational root, \(p/q\), of the polynomial must have \(p\) as a factor of the constant term (15 in this case) and \(q\) as a factor of the leading coefficient (16 in this case). Listing all these possibilities, we have: ±1, ±3, ±5, ±15, ±1/2, ±3/2, ±5/2, ±15/2, ±1/4, ±3/4, ±5/4, ±15/4.

Step 2: Use Synthetic Division to Find a Root

We can use synthetic division to test the potential roots. Start with ±1. If synthetic division results in a remainder of zero, then the tested value is a root. After performing synthetic division, we find that -1 is a root of the function.

Step 3: Polynomial Reduction and Repeat Step 2

After finding a root, -1 in this case, reduce the polynomial by factoring out the term \((x+1)\) using the quotient from synthetic division. The reduced polynomial is \(16x^2-4x-15\). Repeat step 2 on the reduced polynomial to find other roots. We find that the remaining roots are -3/4 and 5/4.

Step 4: Write Polynomial as Product of Linear Factors

Based on the roots found, the polynomial as a product of linear factors can be expressed as: \(f(x) = 16(x+1)(x+\frac{3}{4})(x-\frac{5}{4})\).

Key Concepts

Rational Root Theorem

Understanding the Rational Root Theorem is critical when we want to find the zeros of a polynomial function like \(f(x)=16x^{3}-20x^{2}-4x+15\). This theorem provides a systematic way to list all possible rational zeros of a polynomial function. Simply put, if a polynomial has rational zeros, they must be fractions formed by the divisors of the constant term over the divisors of the leading coefficient.

In our exercise, the constant term is 15, and the leading coefficient is 16. The theorem directs us to consider all fractions where the numerator is a factor of 15 (±1, ±3, ±5, ±15) and the denominator a factor of 16 (±1, ±2, ±4, ±8, ±16), giving us a list of potential rational roots to test using methods like synthetic division or substitution.

Synthetic Division

When we have a list of potential rational roots from the Rational Root Theorem, Synthetic Division is a handy tool for testing these roots without heavy computation. It's a form of long division that simplifies the process of dividing a polynomial by a binomial of the form \((x-c)\).

Let's explore this with our polynomial \(f(x)=16x^{3}-20x^{2}-4x+15\). We found -1 to be a potential root, and when we perform synthetic division with -1, the absence of a remainder confirms that -1 is indeed a root. This successful test allows us to reduce the polynomial to a lower degree, one step closer to expressing it as a product of linear factors.

Factoring Polynomials

Factoring Polynomials is a way of breaking down complex polynomials into simpler components—specifically, products of linear factors and sometimes non-linear factors. After using synthetic division to find a zero of the polynomial, we use the result to rewrite the polynomial.

In the context of our problem, after finding that -1 is a zero, we can factor out \((x+1)\) from the original polynomial. This gives us a reduced quadratic polynomial, which we then further factor or apply the quadratic formula to find the remaining zeros. Properly factoring polynomials is a crucial step towards fully understanding the structure of the polynomial and its roots.

Linear Factors

Polynomials can ultimately be broken down into Linear Factors, which represent its zeros in the simplest form. Once all the zeros of a polynomial are found, the polynomial can be expressed as a product of linear factors, where each factor is in the form of \((x - r)\), with \(r\) being a zero of the polynomial.

For our exercise, after determining all zeros, we rewrite the polynomial \(f(x)=16x^{3}-20x^{2}-4x+15\) in terms of its linear factors: \(f(x) = 16(x+1)(x+\frac{3}{4})(x-\frac{5}{4})\). This not only gives us the roots directly but also offers a straightforward way to evaluate the polynomial at any given value of \(x\).