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Magazine Chapter 2: Problem 59
For each polynomial function given: (a) list each real zero and its multiplicity; (b) determine whether the graph touches or crosses at each \(x\) -intercept; (c) find the \(y\) -intercept and a few points on the graph; (d) determine the end behavior; and (e) sketch the graph. $$f(x)=12 x^{6}-36 x^{5}-48 x^{4}$$
Short Answer
Expert verified
Zeros: \(0(m=4), 4(m=1), -1(m=1)\); Graph touches at 0, crosses at 4 and -1; \(y\)-intercept: 0; End behavior: up on both sides.
Step by step solution
01
Factor the Polynomial
The given function is \( f(x) = 12x^6 - 36x^5 - 48x^4 \). Start by factoring out the greatest common factor, which is \( 12x^4 \). This simplifies the expression to \( f(x) = 12x^4(x^2 - 3x - 4) \). Next, factor \( x^2 - 3x - 4 \) by solving its quadratic equation \( x^2 - 3x - 4 = 0 \). This factors further into \( (x - 4)(x + 1) \). So, the function can be rewritten as \( f(x) = 12x^4(x - 4)(x + 1) \).
02
Find Real Zeros and Their Multiplicities
The factored polynomial \( f(x) = 12x^4(x - 4)(x + 1) \) has zeros at \( x = 0 \), \( x = 4 \), and \( x = -1 \). The zero \( x = 0 \) has a multiplicity of 4, \( x = 4 \) has a multiplicity of 1, and \( x = -1 \) also has a multiplicity of 1.
03
Analyze Graph Behavior at Each Zero
For any zero with even multiplicity, the graph will touch the x-axis and turn around. For any zero with odd multiplicity, the graph will cross the x-axis. Thus, at \( x = 0 \), the graph touches the x-axis, and at \( x = 4 \) and \( x = -1 \), the graph crosses the x-axis.
04
Find the y-Intercept and Some Points
The y-intercept occurs when \( x = 0 \). Substituting into the polynomial gives \( f(0) = 0 \). To find other points, substitute values such as \( x = 1 \) and \( x = -2 \) into the function. For example, \( f(1) = 12(1)^4(1 - 4)(1 + 1) = -72 \) and \( f(-2) = 12(-2)^4((4)(-1)) = 3072 \). Thus, some points are \((0, 0)\), \((1, -72)\), \((-2, 3072)\).
05
Determine End Behavior
For end behavior analysis, observe the leading term of the polynomial. Here, that term is \( 12x^6 \), which is of even degree and has a positive leading coefficient. Hence, as \( x \to \pm \infty \), \( f(x) \to \infty \). This means both ends of the graph will go upwards.
06
Sketch the Graph
Using the identified zeros, behaviors at those zeros, the y-intercept, selected points, and the end behavior, sketch the graph. Starting at the y-intercept \((0, 0)\), the graph flattens out at \(x=0\) due to even multiplicity, crosses the x-axis at \(x=4\) and \(x=-1\), and goes towards infinity as both \(x\) approaches positive or negative infinity.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Real Zeros
Real zeros are the values of \( x \) at which the polynomial equation \( f(x) = 0 \). In simpler terms, they are the points where the graph of the polynomial crosses or touches the x-axis. To find the real zeros of the polynomial function \( f(x) = 12x^6 - 36x^5 - 48x^4 \), we begin by factoring it. The expression factors to \( f(x) = 12x^4(x - 4)(x + 1) \). From this factored form, we can easily identify the real zeros.
- \( x = 0 \) (with a multiplicity of 4)
- \( x = 4 \) (with a multiplicity of 1)
- \( x = -1 \) (with a multiplicity of 1)
Multiplicity
The multiplicity of a zero refers to how many times that particular zero appears as a root in the polynomial's factored form. Understanding multiplicity helps us predict the graph's behavior at each intercept.
- If the multiplicity is even, the graph touches the x-axis and turns around at that intercept without crossing it.
- If the multiplicity is odd, the graph crosses the x-axis at that intercept.
End Behavior
End behavior describes how the values of \( f(x) \) behave as \( x \) approaches very large positive or negative values (\( x \to \pm \infty \)). Observing the leading term of the polynomial, which in this case is \( 12x^6 \), is key for understanding end behavior.Due to its even degree (6) and positive leading coefficient (12):
- As \( x \to -\infty \), \( f(x) \to \infty \)
- As \( x \to \infty \), \( f(x) \to \infty \)
X-Intercepts
An x-intercept is a point where the graph crosses or touches the x-axis, meaning the y-value is zero at these points. By identifying the real zeros of the polynomial \( f(x) = 12x^4(x - 4)(x + 1) \), we can determine its x-intercepts:
- \( (0, 0) \): Touches the axis (since multiplicity is even)
- \( (4, 0) \): Crosses the axis (since multiplicity is odd)
- \( (-1, 0) \): Crosses the axis (since multiplicity is odd)
Y-Intercepts
The y-intercept of a polynomial function is found by evaluating the function at \( x = 0 \). It represents the point where the graph crosses the y-axis. For the polynomial \( f(x) = 12x^4(x - 4)(x + 1) \), substitute \( x = 0 \) to find the y-intercept:
- \( f(0) = 12(0)^4(0 - 4)(0 + 1) = 0 \)
