Problem 233 - Precalculus

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

The t-intercepts are t = 0, t = -5, and t = 2.

Step 1: Identify the Intercepts

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

Step 2: Factor the Polynomial

Factor out the greatest common factor, which is \(4t^2\), to simplify the equation: \[4t^2(t^2 + 3t - 10) = 0\]

Step 3: Solve for t

Since \(4t^2 = 0\) results in \(t = 0\), we now solve the quadratic equation \(t^2 + 3t - 10 = 0\) by factoring. Factoring gives us \((t + 5)(t - 2) = 0\).

Step 4: Find All Solutions

Set each factor equal to zero and solve for \(t\): \[t= 0, \, t+5 = 0 \Rightarrow t = -5, \, t-2 = 0 \Rightarrow t = 2\]

Step 5: Summarize the Intercepts

The t-intercepts of the polynomial function are \(t = 0\), \(t = -5\), and \(t = 2\). These are the values of \(t\) where the polynomial crosses the t-axis.

Key Concepts

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler expressions, called factors, that when multiplied together give the original polynomial. This is especially useful for finding intercepts because it helps simplify equations. In the example given, the polynomial is \(4t^4 + 12t^3 - 40t^2\). The first step is to identify the greatest common factor of all terms, which in this case is \(4t^2\). When we factor \(4t^2\) out, the polynomial becomes \(4t^2(t^2 + 3t - 10)\). By rewriting the expression in this form, we simplify the process of finding roots or solutions to the polynomial equation. The inner term, \(t^2 + 3t - 10\), can be factored further into \((t + 5)(t - 2)\), making it easy to identify additional solutions. Learning to factor polynomials, especially quadratics, involves understanding patterns in numbers and sometimes trial and error. Practicing this skill can greatly aid in understanding how polynomials behave and in solving equations efficiently.

Quadratic Equation

A quadratic equation is a polynomial equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. In the case of the equation we solved, the quadratic portion was \(t^2 + 3t - 10 = 0\). Quadratic equations can often be factored into two binomial expressions, as demonstrated. Once factored, each expression can provide solutions or roots for the equation, which are the values that satisfy it completely. For the equation \(t^2 + 3t - 10\), factoring gives us the expression \((t + 5)(t - 2) = 0\). The solutions to a quadratic can also be found using the quadratic formula \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), but factoring can be quicker when applicable. This algebraic approach allows us to solve for \(t = -5\) and \(t = 2\) directly after factoring, showing the value of being familiar with multiple methods of solving quadratics.

x-Intercepts

Finding \(x\)-intercepts (or \(t\)-intercepts in this problem's context) of a polynomial function involves determining the values where the graph of the function crosses the \(x\)-axis. This is equivalent to setting the polynomial equal to zero and solving for the variable, as values of \(t\) at intercepts make the function's value zero.In the exercise, the polynomial \(C(t) = 4t^4 + 12t^3 - 40t^2\) requires solving for when the entire expression equals zero. After factoring, we solved the resulting equations: \(4t^2 = 0\), \(t + 5 = 0\), and \(t - 2 = 0\). The solutions are \(t = 0\), \(t = -5\), and \(t = 2\), indicating these are the points at which the graph intersects the \(t\)-axis.Recognizing \(x\)-intercepts allows us to understand and analyze graphs of polynomials more effectively. These intercepts provide crucial information about the trend and behavior of polynomial functions and also help in sketching them accurately.