Problem 42 - Algebra for College Students

Question

For Problems \(35-42\), (a) find the \(y\) intercepts, (b) find the \(x\) intercepts, and (c) find the intervals of \(x\) where \(f(x)>0\) and those where \(f(x)<0\). Do not sketch the graphs. $$ f(x)=(x+2)^{5}(x-4)^{2} $$

Step-by-Step Solution

Answer

Y-intercept: \((0, 512)\); X-intercepts: \((-2, 0)\), \((4, 0)\); \(f(x)>0\) for \(x\neq -2, x \neq 4\), \(f(x)<0\) never occurs.

Step 1: Finding the Y-Intercept

To find the y-intercept of the function, we substitute \(x = 0\) into the function \(f(x)\). Thus, \[ f(0) = (0+2)^5(0-4)^2 = 2^5 \times 4^2 = 32 \times 16 = 512. \] Therefore, the y-intercept is at the point \((0, 512)\).

Step 2: Finding the X-Intercepts

To find the x-intercepts, we set \(f(x) = 0\) and solve for \(x\). This gives us: \[ (x+2)^5(x-4)^2 = 0. \] This equation is zero when either \((x+2)^5 = 0\) or \((x-4)^2 = 0\). Solving these, we get \(x = -2\) and \(x = 4\). Hence, the x-intercepts are at the points \((-2, 0)\) and \((4, 0)\).

Step 3: Identifying Intervals where \(f(x) > 0\) and \(f(x) < 0\)

Analyze the behavior of the function's factors for intervals around the roots \(x = -2\) and \(x = 4\). We consider the intervals \((-\infty, -2)\), \((-2, 4)\), and \((4, \infty)\). In \((-\infty, -2)\), both \((x+2)^5\) and \((x-4)^2\) are negative, making \(f(x) > 0\). In \((-2, 4)\), \((x+2)^5\) is positive and \((x-4)^2\) is positive, giving \(f(x) > 0\). In \((4, \infty)\), both factors are positive, hence \(f(x) > 0\). Hence, \(f(x) < 0\) in any interval does not occur here.

Key Concepts

Finding Intercepts

When dealing with polynomial functions, finding intercepts is an essential skill. These intercepts help us determine where the graph crosses the axes. To find the **y-intercept**, substitute zero for every instance of the variable \(x\) in the function. For the function \(f(x) = (x+2)^5(x-4)^2\), this means calculating \(f(0)\):

• Substitute \(0\) into the function: \((0+2)^5(0-4)^2\).
• Simplify each part: \(2^5 = 32\) and \((-4)^2 = 16\).
• Multiply the results: \(32 \times 16 = 512\).

Thus, the point \((0, 512)\) is our y-intercept. The graph of the function touches the y-axis at this point.
For the **x-intercepts**, set the whole function equal to zero and solve for \(x\). This means resolving the equation \((x+2)^5(x-4)^2 = 0\). Since a product of factors equals zero when at least one of the factors is zero, we can find:

• \((x+2)^5 = 0\) leads to \(x = -2\).
• \((x-4)^2 = 0\) results in \(x = 4\).

These calculations give us the x-intercepts at the points \((-2, 0)\) and \((4, 0)\). Hence, the graph crosses the x-axis at these points.

Interval Notation

Interval notation is a way of representing a set of numbers along a number line. For polynomial functions, it's frequently used to describe the set of values where the function is positive or negative. When we talk about intervals, we're referring to every possible \(x\) value within a certain range.
For example, the open interval \((-2, 4)\) includes all numbers between \(-2\) and \(4\), but not \(-2\) and \(4\) themselves. This interval notation is useful because it's concise, making it easy to express large sets of numbers without listing each one.
When you put intervals together with the behavior of functions, you can convey a lot of information. For \(f(x)\), the positive and negative nature of the function can easily be represented in intervals:

• The function \(f(x) = (x+2)^5(x-4)^2\) is positive on \((-\infty, -2)\), \((-2, 4)\), and \((4, \infty)\).
• The function never turns negative within these ranges, thus no negative interval is present.

Whenever you're working with polynomial functions and intervals, make sure to look closely at where the function crosses the x-axis, as these points will create boundary lines for these intervals.

Positive and Negative Intervals

Understanding where a polynomial function is positive or negative helps you understand its behavior across different values of \(x\). This concept is crucial when analyzing functions without graphing them.
To determine these intervals, you need to check the sign of the function in areas divided by its x-intercepts. For \(f(x) = (x+2)^5(x-4)^2\):

• For \((-\infty, -2)\), observe that both \(x+2\)\ and \(x-4\)\ will be negative. However, raising to the fifth and second power ensures that both results on squaring or taking powers makes them positive, hence overall positive.
• In \((-2, 4)\), \(x+2\)\ is always positive and \(x-4\)\ is negative, but when squared, the negative becomes positive. In this interval, the function also stays positive.
• For \((4, \infty)\), since both \(x+2\)\ and \(x-4\)\ are inherently positive for all x-values in this range, the function remains positive.

Knowing these intervals helps predict where a polynomial rises or falls. This assessment doesn't require graphing, saving time when analyzing functions.