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

Step-by-Step Solution

Answer

The polynomial function is \( P(x) = -\frac{1}{4}(x-2)(x-5)(x+3) \).

Step 1: Understand the Zeros and Form the General Polynomial

The zeros of the polynomial are given as 2, 5, and -3. Therefore, the polynomial can be written as:\[ P(x) = a(x-2)(x-5)(x+3) \]where \(a\) is a non-zero real number to be determined.

Step 2: Substitute the Point P(1) = -4

To find the value of \(a\), use the condition that \(P(1) = -4\). Substitute \(x = 1\) into the polynomial:\[ -4 = a(1-2)(1-5)(1+3) \]

Step 3: Simplify the Equation to Solve for a

Simplify the expression in the equation:\[ -4 = a(-1)(-4)(4) \]\[ -4 = a imes 16 \]

Step 4: Solve for a

Solve for \(a\) by dividing both sides by 16:\[ a = \frac{-4}{16} = -\frac{1}{4} \]

Step 5: Write the Final Polynomial Function

Substitute \(a = -\frac{1}{4}\) back into the polynomial:\[ P(x) = -\frac{1}{4}(x-2)(x-5)(x+3) \]

Key Concepts

Understanding the Degree of a Polynomial

The degree of a polynomial is one of its most important characteristics. It tells us the highest power of the variable present in the polynomial expression. For a polynomial function \(P(x)\) of degree 3, this means the highest exponent of \(x\) is 3.

• Example: For \(P(x) = a(x-2)(x-5)(x+3)\), the degree is 3 because there are three linear factors, each contributing a degree of 1.
• The number of factors (when fully factored) typically equals the degree if all factors are of degree 1.

Understanding the degree helps to predict the general shape of the polynomial's graph and can indicate the number of roots or zeros the function will have.

Real Coefficients in Polynomials

Polynomials with real coefficients have coefficients that are real numbers, meaning they can be whole numbers, fractions, or even decimals. The function defined by these coefficients behaves predictably when plotted on a real plane. Real coefficients ensure that if the polynomial has complex roots, those roots will occur in conjugate pairs.

• In the polynomial \(P(x) = -\frac{1}{4}(x-2)(x-5)(x+3)\), the coefficient \(-\frac{1}{4}\) is a real number.
• Real coefficients make the function applicable in real-life scenarios, as real numbers are observable and measurable.

The predictability of polynomials with real coefficients is one of the reasons they are widely used in scientific and engineering calculations.

Exploring the Zeros of a Polynomial

Zeros of a polynomial, also known as roots or solutions, are values of \(x\) that make the polynomial equal to zero. In this context, zeros are solutions to the equation \(P(x) = 0\). For the polynomial of degree 3 given in the exercise, zeros are 2, 5, and -3.

• Each zero corresponds to a factor in the polynomial: 2 corresponds to \((x-2)\), 5 to \((x-5)\), and -3 to \((x+3)\).
• Zeros help in constructing the polynomial, as each zero appears as a factor.

Finding zeros also allows you to sketch the graph of the polynomial, as these points will cross the \(x\)-axis. Understanding zeros is crucial for solving polynomial equations.

Solving for Coefficients in Polynomials

Solving for coefficients involves determining the specific values that multiply each term in the polynomial. This process often requires substituting given points into the polynomial to find unknown coefficients. In the exercise, determining the leading coefficient \(a\) was essential.

• Use known conditions, such as \(P(1) = -4\), to find \(a\).
• Substitute \(x = 1\) into the polynomial equation, setting \(P(1) = -4\).
• Solve \(-4 = a(-1)(-4)(4)\) and find \(a = -\frac{1}{4}\).

Once the coefficient is found, it is integrated back into the polynomial expression, completing the polynomial function. This technique is often used in real-life applications where conditions or data points are known.