Problem 10 - College Algebra with Corequisite Support

Question

For the following exercises, find the \(x\) - or t-intercepts of the polynomial functions. $$ C(t)=2 t^{4}-8 t^{3}+6 t^{2} $$

Step-by-Step Solution

Answer

The intercepts are at \( t = 0, 1, 3 \).

Step 1: Set the function equal to zero

To find the intercepts, we first set the polynomial function equal to zero. This means we solve for \( t \) in the equation: \[ 2t^4 - 8t^3 + 6t^2 = 0. \]

Step 2: Factor out the greatest common factor

Next, look for and factor out the greatest common factor from all terms. Here, the greatest common factor of each term is \( 2t^2 \). Factoring \( 2t^2 \) out gives: \[ 2t^2(t^2 - 4t + 3) = 0. \]

Step 3: Solve each factor for zero

We apply the zero product property, which states if \( a \cdot b = 0 \), then either \( a = 0 \) or \( b = 0 \).1. First, solve \( 2t^2 = 0 \): \[ t^2 = 0 \] \[ t = 0. \] 2. Second, solve the quadratic \( t^2 - 4t + 3 = 0 \) using factoring or the quadratic formula. Notice this can be factored into: \[ (t-1)(t-3) = 0. \] So, the solutions are: \( t = 1 \) and \( t = 3. \)

Key Concepts

x-intercepts

To understand x-intercepts, imagine a graph of a polynomial function. The x-intercepts are the points where the graph crosses the x-axis. In other words, these are the values of the variable where the function equals zero. Finding these points is crucial, as they reveal important information about the behavior of the function.
In mathematical exercises, to find x-intercepts, we set the function equal to zero and solve for the variable involved. For instance, given a polynomial function like in the original exercise, we solve for the variable by setting the equation equal to zero:

• Set the function to zero: \[C(t)=2t^4 - 8t^3 + 6t^2 = 0\]
• Solve the equation for the variable (in this case, \(t\)).

This approach leads us to find the values of \(t\) where the graph of \(C(t)\) intersects the t-axis, revealing these intercepts.

factoring polynomials

Factoring polynomials is like taking apart a complex structure to reveal its simpler building blocks. It involves breaking down a polynomial equation into the product of simpler expressions, or factors, that are easier to solve.
For the polynomial function in our example, we first look for a greatest common factor (GCF) to simplify the expression. Here, the GCF in \[2t^4 - 8t^3 + 6t^2\] is \(2t^2\). Factoring this out transforms the equation into:

• First factor: \(2t^2(t^2 - 4t + 3) = 0\)

Breaking down further, we might need to factor the remaining quadratic expression if possible. Knowing how to factor polynomials not only simplifies finding solutions but also provides deeper insights into the nature of the polynomial itself.

zero product property

The zero product property is a key principle when working with polynomials, particularly after factoring. This property tells us that if a product of two expressions equals zero, then each expression must be zero. It is pivotal for finding the x-intercepts once a polynomial is factored.
In our exercise, once we have the factored form as:\[ 2t^2(t^2 - 4t + 3) = 0 \] we apply the zero product property:

• First equation: \(2t^2 = 0\). Solving gives us \(t = 0\).
• Second equation from the quadratic: \((t - 1)(t - 3) = 0\). Solving provides \(t = 1\) and \(t = 3\).

This step identifies all possible solutions, clearly identifying where the t-intercepts are and showing where the polynomial equals zero. The zero product property is a powerful tool in algebra that directly connects factors to solutions.