Problem 11 - College Algebra

Question

For the following exercises, find the \(x\) - or \(t\) -intercepts of the polynomial functions. $$ C(t)=4 t^{4}+12 t^{3}-40 t^{2} $$

Step-by-Step Solution

Answer

t-intercepts: 0, 2, -5.

Step 1: Set the Polynomial Equal to Zero

To find the t-intercepts, we need to set the polynomial function equal to zero: \[ C(t) = 4t^4 + 12t^3 - 40t^2 = 0 \] and solve for \( t \).

Step 2: Factor Out the Greatest Common Factor

Each term in \( 4t^4 + 12t^3 - 40t^2 \) has a common factor \( 4t^2 \). Factor out \( 4t^2 \): \[ 4t^2(t^2 + 3t - 10) = 0 \].

Step 3: Solve for t from the Factored Equation

We have two factors: \( 4t^2 = 0 \) and \( t^2 + 3t - 10 = 0 \). Solve \( 4t^2 = 0 \):- \( t^2 = 0 \) so \( t = 0 \).Now solve \( t^2 + 3t - 10 = 0 \) using the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = 3, c = -10 \).

Step 4: Apply the Quadratic Formula

Using the quadratic formula: \[ t = \frac{-3 \pm \sqrt{3^2 - 4 \times 1 \times (-10)}}{2 \times 1} \] \[ t = \frac{-3 \pm \sqrt{9 + 40}}{2} \] \[ t = \frac{-3 \pm \sqrt{49}}{2} \] \[ t = \frac{-3 \pm 7}{2} \]This gives two solutions: \( t = \frac{4}{2} = 2 \) and \( t = \frac{-10}{2} = -5 \).

Step 5: List the Intercepts

We have found the following t-intercepts for the polynomial: \( t = 0 \), \( t = 2 \), and \( t = -5 \).

Key Concepts

Factoring Polynomials

Factoring polynomials is a method that involves rewriting a polynomial as a product of simpler polynomials. This helps in solving polynomial equations, particularly when finding roots or intercepts. The process begins by identifying common factors in all terms of the polynomial.

In the given polynomial function \(C(t) = 4t^4 + 12t^3 - 40t^2\), the first step in factoring is to find and pull out the greatest common factor (GCF). Here, the GCF is \(4t^2\), since each term includes this factor.

• Take out \(4t^2\) to simplify the expression
• Remain with \(t^2 + 3t - 10\) as a second factor

Once factored, the polynomial translates into \(4t^2(t^2 + 3t - 10) = 0\). This step is crucial as it breaks down the complexity of the expression, allowing for easier solutions to find the polynomial's intercepts.

Quadratic Formula

The quadratic formula is a powerful tool used for finding the roots of a quadratic equation, which has the standard form \(ax^2 + bx + c = 0\). It is particularly useful when factoring becomes difficult or inefficient.

In the polynomial \(t^2 + 3t - 10 = 0\), it is quadratic with coefficients: \(a = 1\), \(b = 3\), and \(c = -10\). The quadratic formula is applied as follows:

• Insert the coefficients into the formula: \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
• Compute: \(t = \frac{-3 \pm \sqrt{3^2 - 4 \times 1 \times (-10)}}{2 \times 1}\)
• Simplify under the square root: \(t = \frac{-3 \pm \sqrt{49}}{2}\)
• This results in \(t = 2\) and \(t = -5\)

These solutions give us the roots, which in our problem are the intercepts \(t = 2\) and \(t = -5\). Mastery of this formula is essential for handling more complex polynomial expressions.

Polynomial Functions

Polynomial functions are mathematical expressions involving a variable raised to different powers, combined using addition, subtraction, and multiplication. Understanding the nature of polynomial functions is vital for solving equations and analyzing curves.

For example, the polynomial function given by \(C(t) = 4t^4 + 12t^3 - 40t^2\) is a fourth-degree polynomial, indicating it can have up to four real roots or solutions. Not every polynomial will be easy to factor or solve manually, making it necessary to learn various methods like factoring and using the quadratic formula.

Polynomial functions often represent physical phenomena, such as motion, growth patterns, or changes over time. As such, finding their intercepts, as done in this exercise, helps build a deeper understanding of their behavior and underlying processes.