Problem 66 - College Algebra

Question

For the following exercises, construct a polynomial function of least degree possible using the given information. Real roots: \(-1,1,3\) and \((2, f(2))=(2,4)\)

Step-by-Step Solution

Answer

The polynomial is \( p(x) = -\frac{4}{3}(x + 1)(x - 1)(x - 3) \).

Step 1: Identify the Roots and Factors

Given the real roots \(-1, 1, 3\), each root corresponds to a factor of the polynomial. Thus, the factors are \((x + 1)\), \((x - 1)\), and \((x - 3)\).

Step 2: Construct Polynomial from Factors

Since these are all linear factors and given as the roots of the polynomial, the polynomial can be written as the product \(p(x) = a(x + 1)(x - 1)(x - 3)\), where \(a\) is a constant to be determined.

Step 3: Determine the Leading Coefficient

To find the value of \(a\), use the condition \(f(2) = 4\). Substitute into \(p(x)\): \(p(2) = a(2 + 1)(2 - 1)(2 - 3) = 4\). This gives us \(a(3)(1)(-1) = 4\).

Step 4: Solve for \(a\)

From \(a(3)(1)(-1) = 4\), we get \(-3a = 4\). Solving for \(a\), we find \(a = -\frac{4}{3}\).

Step 5: Write the Full Polynomial

The polynomial of least degree is \(p(x) = -\frac{4}{3}(x + 1)(x - 1)(x - 3)\). To expand, distribute the factors: first combine \((x+1)(x-1) = x^2 - 1\), then \((x^2 - 1)(x-3) = x^3 - 3x^2 - x + 3\). Multiply by \(-\frac{4}{3}\) to get the expanded polynomial \(p(x) = -\frac{4}{3}(x^3 - 3x^2 - x + 3)\).

Key Concepts

Real Roots

Real roots of a polynomial function are the values of \(x\) that make the polynomial equal to zero. These roots are the x-values where the graph of the polynomial crosses or touches the x-axis. In the context of the exercise, the given roots are \(-1\), \(1\), and \(3\). Real roots are crucial in constructing the polynomial since each root corresponds to a factor of the polynomial. For each root \(r\), there is a corresponding factor of the form \((x - r)\). So, for roots \(-1\), \(1\), and \(3\), we have the factors \((x + 1)\), \((x - 1)\), and \((x - 3)\) respectively.
Understanding real roots and their associated factors helps us form the polynomial expression efficiently.

Leading Coefficient

The leading coefficient in a polynomial is the coefficient of the term with the highest degree. It significantly influences the end behavior of the polynomial graph. In simple terms, it dictates how the graph of the polynomial opens with respect to the x-axis when extended to infinity on either side.

In our exercise, the polynomial is initially written in the factored form \(p(x) = a(x + 1)(x - 1)(x - 3)\). The term \(a\) is the leading coefficient, which needs to be determined. To find \(a\), a specified condition is used: \(f(2) = 4\). By substituting \(x = 2\) into the polynomial and solving, \(a\) is determined to be \(-\frac{4}{3}\). With \(a\) known, the leading coefficient in the expanded form \(-\frac{4}{3}(x^3 - 3x^2 - x + 3)\) becomes \(-\frac{4}{3}\).
Realizing the role of the leading coefficient helps in understanding both the algebra and geometry of polynomial functions.

Factored Form

Factored form of a polynomial is the expression of the polynomial as a product of its factors. This form is particularly useful for identifying the polynomial's roots directly and for seeing its basic shape without fully expanding it.
The benefit of using the factored form is its ability to show us the zeros (roots) of the function right away. In the given exercise, the polynomial starts as \(p(x) = a(x + 1)(x - 1)(x - 3)\). Here, \((x + 1)\), \((x - 1)\), and \((x - 3)\) reveal the roots \(-1\), \(1\), and \(3\), respectively. Factored form is also instrumental in determining if a graph crosses or merely touches the x-axis at given points.
The impact of the factor form simplifies initial understanding and solving of polynomial functions.

Polynomial Degree

The degree of a polynomial is the highest power of \(x\) when the polynomial is expressed in its expanded form. It is a core characteristic that tells you about the polynomial's overall shape and the maximum number of real roots it could have, assuming all coefficients are real.
In this exercise, the polynomial \(p(x) = -\frac{4}{3}(x + 1)(x - 1)(x - 3)\) is a third-degree polynomial. This can be seen when the polynomial is fully expanded into \(-\frac{4}{3}(x^3 - 3x^2 - x + 3)\), where the highest power of \(x\) is 3. A polynomial of degree 3 will have at most three real roots, which aligns with the given roots: \(-1\), \(1\), and \(3\).
Grasping the polynomial degree helps us predict the number of roots and shape of its graph quickly.