Problem 10 - 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 \(2,5,\) and \(-3 ; \quad P(1)=-4\)

Step-by-Step Solution

Answer

The polynomial is \( P(x) = -\frac{1}{4}x^3 + x^2 + \frac{11}{4}x - \frac{15}{2} \).

Step 1: Express the Polynomial in Factored Form

Since the given polynomial has zeros at 2, 5, and -3, we know that it can be written in the factored form: \( P(x) = a(x-2)(x-5)(x+3) \), where \( a \) is a constant to be determined.

Step 2: Use Given Point to Find 'a'

We know from the problem that \( P(1) = -4 \). Substitute 1 into the polynomial: \( -4 = a(1-2)(1-5)(1+3) \). This simplifies to \( -4 = a(-1)(-4)(4) \).

Step 3: Solve for 'a'

Calculate the expression: \(-4 = 16a\). Divide both sides by 16 to find \( a \): \( a = \frac{-4}{16} = -\frac{1}{4} \).

Step 4: Write the Polynomial in Standard Form

Substitute \( a = -\frac{1}{4} \) back into the polynomial: \( P(x) = -\frac{1}{4}(x-2)(x-5)(x+3) \). Expand the expression to its standard form, combining terms and simplifying: First, expand \((x-2)(x-5)\): \\((x-2)(x-5) = x^2 - 5x - 2x + 10 = x^2 - 7x + 10\).Next, multiply by \((x+3)\): \[ (x^2 - 7x + 10)(x + 3) = x^3 + 3x^2 - 7x^2 - 21x + 10x + 30 = x^3 - 4x^2 - 11x + 30 \]Now multiply by \(-\frac{1}{4}\) as determined earlier: \[ P(x) = -\frac{1}{4}(x^3 - 4x^2 - 11x + 30) = -\frac{1}{4}x^3 + x^2 + \frac{11}{4}x - \frac{30}{4} \].Simplify further: \[ P(x) = -\frac{1}{4}x^3 + x^2 + \frac{11}{4}x - \frac{15}{2} \].

Key Concepts

Factored Form

The factored form of a polynomial is a way of expressing the polynomial as a product of its factors. In the case of the polynomial function given in the exercise, the zeros or roots were provided: 2, 5, and -3. These zeros indicate where the polynomial function intersects the x-axis. By using these zeros, the polynomial can be expressed in the factored form as:

• For the zero at 2, the factor is \(x - 2\).
• For the zero at 5, the factor is \(x - 5\).
• For the zero at -3, the factor is \(x + 3\).

Thus, we can write the polynomial as \(P(x) = a(x-2)(x-5)(x+3)\), where \(a\) is a constant that can be determined by additional conditions, such as a given point through which the polynomial passes. The factored form is particularly useful for quickly identifying the roots of a polynomial.

Polynomial Roots

The roots or zeros of a polynomial are the values of \(x\) that make the polynomial equal to zero. These are critical because they reveal where the graph of the polynomial function crosses the x-axis.Given the zeros 2, 5, and -3, these are the solutions to the equations:

• \(x - 2 = 0\)
• \(x - 5 = 0\)
• \(x + 3 = 0\)

Each equation corresponds to one root of the polynomial.
To find the complete polynomial function, it is important to express it in terms of these roots, often through the factored form. Consequently, the polynomial can be expressed as a product of these factors multiplied by a constant.
In the exercise, the next step was to find the value of \(a\) using a given condition, \(P(1) = -4\). This helps in customizing the polynomial to fit specific conditions.

Standard Form

The standard form of a polynomial is a way of writing it as a sum rather than as a product. It is the expanded form of the polynomial that lists all terms in descending order of power.From the factored form \(P(x) = -\frac{1}{4}(x-2)(x-5)(x+3)\), we expanded it to find the standard form. Expanding means carrying out the multiplication:

• First, calculate the multiplication of two factors, such as \((x-2)(x-5)\) to get \((x^2 - 7x + 10)\).
• Next, multiply the result by \((x+3)\) to obtain \((x^3 - 4x^2 - 11x + 30)\).
• Finally, multiply by the constant \(-\frac{1}{4}\) to achieve \(-\frac{1}{4}x^3 + x^2 + \frac{11}{4}x - \frac{15}{2}\).

The standard form is excellent for identifying the degree of the polynomial and for evaluating it at various points. It's more straightforward to work with when graphing or when fitting the polynomial to specific conditions, illustrating both the overall shape and details of the polynomial's behavior.