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For the polynomial $y=-\frac{1}{2} x(4 x+1)^4(x-8)^2$, find the following:


AThe end behavior of the graph.
  • $\lim _{x \rightarrow-\infty} f(x)=\infty$ (the function rises toward infinity on the left), and $\lim _{x \rightarrow \infty} f(x)=-\infty$ (the function falls toward negative infinity on the right)
  • $\lim _{x \rightarrow-\infty} f(x)=-\infty$ and $\lim _{x \rightarrow \infty} f(x)=-\infty$ (the function falls toward negative infinity on both the left and right).
  • $\lim _{x \rightarrow-\infty} f(x)=\infty$ and $\lim _{x \rightarrow \infty} f(x)=\infty$ (the function rises toward infinity on both the left and right).
  • $\lim _{x \rightarrow-\infty} f(x)=-\infty$ (the function falls toward negative infinity on the left), and $\lim _{x \rightarrow \infty} f(x)=\infty$ (the function rises toward infinity on the right).



When listing $x$-intercepts, give $x$-values only separated by commas. Round decimals to 1 decimal place.


BList all $x$-intercepts (or state "No $x$-intercepts").


CList all $x$-intercepts where the graph crosses the $x$-axis (or state "No $x$-intercepts where graph crosses $x$-axis").


DList all $x$-intercepts where the graph touches the $x$-axis and turns (or state "No $x$-intercepts where graph touches $x$-axis and turns").


EThe $y$-intercept (give the $y$-value only).

Verified Answer :

  • The polynomial has a negative leading coefficient and an odd degree of 7, so the end behavior is: \(\lim _{x \rightarrow-\infty} f(x)=\infty\) and \(\lim _{x \rightarrow \infty} f(x)=-\infty\).
  • The \(x\)-intercepts are \(0, -0.3, 8\).
  • The graph crosses the \(x\)-axis at \(0\).
  • The graph touches and turns at \(-0.3, 8\).
  • The \(y\)-intercept is \(0\).
  • Final Answer: \(\boxed{a: \lim _{x \rightarrow-\infty} f(x)=\infty, \lim _{x \rightarrow \infty} f(x)=-\infty; b: 0, -0.3, 8; c: 0; d: -0.3, 8; e: 0}\)

Explanation

  1. Analyzing Polynomial Characteristics and End Behavior

We are given the polynomial \(f(x) = -\frac{1}{2} x(4x+1)^4(x-8)^2\). First, we identify its degree and leading coefficient to determine its end behavior. The degree of the polynomial is the sum of the exponents of its linear factors: \(1 + 4 + 2 = 7\). The leading term can be found by multiplying the leading terms of each factor: \(-\frac{1}{2} \cdot x \cdot (4x)^4 \cdot x^2 = -\frac{1}{2} \cdot x \cdot 256x^4 \cdot x^2 = -128x^7\). Since the degree is odd (\(7\)) and the leading coefficient is negative (\(-128\)), as \(x \rightarrow -\infty\), \(f(x) \rightarrow \infty\), and as \(x \rightarrow \infty\), \(f(x) \rightarrow -\infty\).

  1. Determining x-intercepts

Next, we determine the \(x\)-intercepts by setting \(f(x) = 0\):

  • \(x = 0\)
  • \(4x + 1 = 0 \Rightarrow x = -0.25\)
  • \(x - 8 = 0 \Rightarrow x = 8\)

Rounding these to one decimal place, the intercepts are \(0.0, -0.3,\) and \(8.0\). The \(x\)-intercepts are \(0, -0.3, 8\).

  1. Determining Crossing and Turning Behavior and y-intercept

The behavior at each \(x\)-intercept depends on the multiplicity of its corresponding factor:

  • For \(x=0\), the factor is \(x^1\). Since the multiplicity is 1 (odd), the graph crosses the \(x\)-axis at \(x=0\).
  • For \(x=-0.25\), the factor is \((4x+1)^4\). Since the multiplicity is 4 (even), the graph touches the \(x\)-axis and turns at \(x \approx -0.3\).
  • For \(x=8\), the factor is \((x-8)^2\). Since the multiplicity is 2 (even), the graph touches the \(x\)-axis and turns at \(x=8\).

The \(y\)-intercept is calculated by evaluating \(f(0) = -\frac{1}{2} \cdot 0 \cdot (4(0)+1)^4 \cdot (0-8)^2 = 0\).

Examples

Polynomial analysis is essential in engineering and physical sciences. For instance, in structural engineering, the bending moment of a beam can often be represented by a polynomial function. By analyzing the intercepts and end behavior of these functions, engineers can determine points of zero stress (intercepts) and how the structure behaves under extreme loads (end behavior), ensuring the safety and stability of buildings and bridges.

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