Problem 7 - A Graphical Approach to Precalculus with Limits

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 \(P(x) = -\frac{1}{6}(x + 3)(x + 1)(x - 4)\).

Step 1: Identify the Form of the Polynomial

Since the polynomial function has zeros at \(-3, -1,\) and \(4\), we can write the polynomial in its factored form: \[ P(x) = a(x + 3)(x + 1)(x - 4) \]Here, \(a\) is a constant that we need to determine using additional conditions provided.

Step 2: Substitute the Extra Condition

We know from the problem statement that \(P(2) = 5\). Substitute \(x = 2\) into the factored form of the polynomial and set it equal to 5:\[ P(2) = a(2 + 3)(2 + 1)(2 - 4) = 5 \]

Step 3: Simplify and Solve for the Constant

Calculate each term:\(2 + 3 = 5,\)\(2 + 1 = 3,\)\(2 - 4 = -2\).Substitute these values into the equation:\[ a \cdot 5 \cdot 3 \cdot (-2) = 5 \]This simplifies to:\[ -30a = 5 \]Solve for \(a\):\[ a = -\frac{1}{6} \]

Step 4: Write the Full Polynomial

Now that we know \(a = -\frac{1}{6}\), substitute it back into the factored form: \[ P(x) = -\frac{1}{6}(x + 3)(x + 1)(x - 4) \] This is the polynomial function of degree 3 with real coefficients satisfying the given conditions.

Key Concepts

Zeros of Polynomial

A critical concept in understanding polynomial functions is the idea of zeros. A zero of a polynomial is a value of \(x\) for which the polynomial evaluates to zero. Simply put, it's a spot where the graph of the polynomial intersects the x-axis.
For a polynomial of degree 3, like the one in our problem, there can be up to three zeros, and these can be real or complex numbers.
If all zeros are real, the polynomial can be expressed in factored form, using its zeros. In our case, the polynomial has real zeros at \(-3, -1, \) and \(4\). This means:

• Each zero can be converted into a factor: \((x + 3), (x + 1),\) and \((x - 4)\).
• The entire polynomial can be built by multiplying these factors together.

This expression helps us quickly see the points where the polynomial graph crosses the x-axis.

Degree of Polynomial

The degree of a polynomial refers to the highest power of \(x\) present in the expression. This is a key feature because it hints at the general shape of the graph, the number of intersecting points, and the behavior of the polynomial at extremes.
For example, a degree 3 polynomial means that the highest power of \(x\) will be \(x^3\).
This also predicts the following properties of the polynomial:

• The graph will have up to three zeros.
• The end behavior: as \(x\) approaches infinity or negative infinity, the graph will tend to infinity or negative infinity, depending on the leading coefficient.

In our exercise, the polynomial is specifically of degree 3, thus it has a cubic form and behaves according to the rules of degree 3 polynomials.

Factored Form

The factored form of a polynomial is essential for easily identifying its zeros. It’s simply a way of writing the polynomial as a product of its factors.
For example, if a polynomial has zeros at \(-3, -1,\) and \(4\), in factored form it becomes \((x + 3)(x + 1)(x - 4)\).
Having the polynomial in this form has several advantages:

• We can easily pinpoint the zeros by setting the factored form equal to zero.
• It simplifies determining certain values of the polynomial, like initial or boundary conditions, by straightforward substitution of \(x\).
• It can be expanded numerically or left in factored form for more analytical approaches.

In our exercise, determining the constant \(a\) when given additional conditions, like \(P(2) = 5\), is remarkably simplified using its factored form.

Real Coefficients

When polynomials have real coefficients, it means all the numbers in front of the variables are real numbers, not complex ones. This includes the constant term \(a\) as well as the numbers you see in the expanded form of the polynomial.
Real coefficients give certain properties to the polynomial:

• If all the coefficients are real, the zeros can either be real or exist in conjugate pairs if they are complex.
• Knowing the coefficients are real helps while multiplying terms in factored form, ensuring all terms in the expanded polynomial remain real.

In the problem presented, every element including the constant \(a = -\frac{1}{6}\) is a real number, illustrating how even though the factor terms might be negative, the coefficients themselves remain real.