Problem 7 - 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. Zeros of \(-3,-1,\) and \(4 ; \quad P(2)=5\)

Step-by-Step Solution

Answer

The polynomial is \(P(x) = -\frac{5}{28}(x^3 - 12x - 12)\).

Step 1: Understand zeros and polynomial structure

Given the zeros are \(-3, -1,\) and \(4\), the polynomial can start as \(P(x) = a(x + 3)(x + 1)(x - 4)\). Here, \(a\) is a constant that ensures the polynomial passes through the given point \(P(2)=5\).

Step 2: Expand the polynomial

First, expand \((x + 3)(x + 1)\) to \(x^2 + 4x + 3\). Next, multiply the result by \((x - 4)\):\[(x^2 + 4x + 3)(x - 4) = x^3 - 4x^2 + 4x^2 - 16x + 3x - 12 = x^3 - 12x - 12.\] So, the polynomial is \(P(x) = a(x^3 - 12x - 12)\).

Step 3: Use the given condition to find \(a\)

We are given that \(P(2) = 5\), substitute \(x = 2\) in the polynomial and set it equal to 5: \(a(2^3 - 12\times2 - 12) = 5\). So, \(a(8 - 24 - 12) = 5\). Thus \(a(-28) = 5\) which implies \(a = -\frac{5}{28}\).

Step 4: Write the final polynomial

Substitute \(a = -\frac{5}{28}\) back into the polynomial \(P(x) = a(x^3 - 12x - 12)\) to get: \(P(x) = -\frac{5}{28}(x^3 - 12x - 12)\).

Key Concepts

Zeros of a Polynomial

Zeros of a polynomial, also called roots, are the values of the variable that make the polynomial equal to zero. In simpler terms, when you plug these numbers into the polynomial instead of the variable, the result is zero. For a polynomial of degree 3, such as the one given in our exercise, there can be up to three zeros. These zeros might be real numbers, complex numbers, or even a combination of both.

Understanding zeros helps in constructing the polynomial function. If a polynomial has zeros at

• -3, -1, and 4

, you know that

• (x+3)(x+1)(x-4)

are factors of that polynomial. This concept ties closely with the Fundamental Theorem of Algebra, which assures us that a polynomial will always have a number of zeros equivalent to its degree over complex numbers. This knowledge aids in forming the polynomial equation more intuitively.

Expanding Polynomials

Expanding polynomials involves expressing the product of factors as a sum of different powers of a variable. This is necessary to simplify the polynomial for further operations. In the given problem, the polynomial wasn’t completely expanded in the beginning. Starting with

• (x + 3)(x + 1)(x - 4)

, the aim was to multiply these factors to achieve a standard polynomial expression.

When expanding, it involves a systematic distribution; first, the binomials

• (x + 3)(x + 1)

are expanded to give

• x^2 + 4x + 3

. Next, multiply the resulting expression with the remaining binomial

• (x - 4)

, to result in the expanded form:

• x^3 - 12x - 12

. This process helps one clearly see all the terms involved, which is crucial for solving or simplifying polynomial equations.

Real Coefficients

Polynomials with real coefficients are those in which every constant term in the polynomial expression is a real number. Real numbers include positive and negative integers, fractions, and irrational numbers, such as

• \(\sqrt{2}\)

, among others. This is important because having real coefficients implies that all arithmetic involving these polynomials strictly utilizes real-number arithmetic.

In the exercise, we're explicitly stated to find a polynomial with real coefficients. This means that while the zeros might appear complex in some cases, all parts of the expanded polynomial after inserting any zero and multiplying should simplify into real numbers. It ensures that the polynomial remains real and applicable in real-world scenarios, where most measurements and data are expressed in real numbers.

Cubic Polynomials

Cubic polynomials are polynomial expressions of degree 3. Their general form is

• ax^3 + bx^2 + cx + d

. They are called cubic because the highest power of the variable is three, resembling a cube when considering volumes in geometry. They can have up to three real zeros or mix real and complex zeros within the same expression.

In the given problem, the task is to form a cubic polynomial. This meant working with terms up to the power of three, effectively combining the three roots

• -3, -1, and 4

into a compact polynomial form. Cubic polynomials are fascinating because they can curve and twist in various ways, making them highly flexible in modeling real-world scenarios like growth trends, physics phenomena, and other complex relationships. Understanding this flexibility and their structure empowers students to better model and analyze various scientific and mathematical problems.