Problem 6 - College Algebra

Question

For the following exercises, find the \(x\) - or \(t\) -intercepts of the polynomial functions. $$ C(t)=2(t-4)(t+1)(t-6) $$

Step-by-Step Solution

Answer

The t-intercepts are at \(t = 4\), \(t = -1\), and \(t = 6\).

Step 1: Understanding Intercepts

Intercepts are the points where the graph of a polynomial crosses the axes. An intercept with the t-axis (or x-axis) occurs when the value of the function is zero. That is, to find intercepts of the polynomial \( C(t) \), we need to solve the equation \( C(t) = 0 \).

Step 2: Setting the Polynomial to Zero

The polynomial given is \( C(t) = 2(t-4)(t+1)(t-6) \). To find the t-intercepts, set \( C(t) = 0 \). Thus, we have the equation: \[ 2(t-4)(t+1)(t-6) = 0 \] This means that at least one of the factors must be zero.

Step 3: Solving Each Factor for Zero

The equation \[ 2(t-4)(t+1)(t-6) = 0 \]implies that \((t-4) = 0\), \((t+1) = 0\), or \((t-6) = 0\). Solve each of these equations individually.1. \(t-4 = 0\) gives \(t = 4\).2. \(t+1 = 0\) gives \(t = -1\).3. \(t-6 = 0\) gives \(t = 6\).

Step 4: Identifying the Intercepts

Now that we have solved the equations, the intercepts where the polynomial function intersects the t-axis are at \(t = 4\), \(t = -1\), and \(t = 6\). These are the points (4,0), (-1,0), and (6,0) on the graph. The factor \(2\) does not affect the intercept values as it is a scalar multiple, which does not influence when the polynomial equals zero.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions that involve summing multiples of variables, raised to varying powers, all added together. The general form of a polynomial function is expressed as follows:

• Third-order: \( f(x) = ax^3 + bx^2 + cx + d \)
• Bear in mind that each term involves variables raised to integer powers (exponents).
• The highest power dictates the "degree" of the polynomial, such as "quadratic" for degree 2 and "cubic" for degree 3.
• Polynomials are often categorized by their degree, which influences their behavior and shape.

In the case of the function given in our exercise, \( C(t) = 2(t-4)(t+1)(t-6) \), we have a third-degree polynomial. This tells us it is cubic and will graphically appear with one bend or fold. Understanding polynomials will help when learning about roots, intercepts, and the overall behavior of graphs.

Intercepts

Intercepts are crucial points where the graph of a polynomial function meets the axes. There are two types:

• X-intercepts (or t-intercepts): Points where the graph crosses the x-axis. Here, the output value is zero \(y=0\).
• Y-intercepts: Points where the graph crosses the y-axis. Here, the input value is zero \(x=0\).

For the polynomial \( C(t) = 2(t-4)(t+1)(t-6) \), finding t-intercepts involves setting \( C(t) = 0 \). By solving this equation, you discover that the function has t-intercepts at \( t = 4, -1, \text{and } 6 \). This gives us the graphical points \((4,0)\), \((-1,0)\), and \((6,0)\). An intercept is essentially where the function crosses the t-axis, showcasing its roots.

Factoring Polynomials

Factoring polynomials is a method of expressing a polynomial as a product of simpler polynomials. It is crucial for finding the roots or intercepts of the polynomial. Here’s how it works:

• Consider the given polynomial \( C(t) = 2(t-4)(t+1)(t-6) \).
• This expression is already factored, presented as a product of binomials and a scalar.
• Each factor corresponds to a root of the polynomial, indicating where it will intersect with the x-axis.
• For any polynomial \( f(x) = 0 \), if \( (x-a) \) is a factor then \( x = a \) is a root or zero of the polynomial.

By solving \( (t-4)(t+1)(t-6) = 0 \), you determine t-values that make the expression zero. This convenient structure allows easy computation of intercepts, a process particularly useful in graphing. Factoring reveals essential properties, like symmetry and turning points.