Problem 64 For each polynomial function giv... [FREE SOLUTION]

Chapter 2: Problem 64

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. $$f(x)=x^{3}-x^{2}-x+1$$

Short Answer

Expert verified

The real zeros are $x = 1$ (multiplicity 2, touches) and $x = -1$ (multiplicity 1, crosses). The y-intercept is $(0,1)$. End behavior: $x \to -\infty, f(x) \to -\infty; x \to +\infty, f(x) \to +\infty$. Plot the intercepts and other points for the graph.

Step by step solution

Finding the Real Zeros

To find the zeros of the polynomial function $f(x) = x^3 - x^2 - x + 1$, we set $f(x)$ equal to zero:\[x^3 - x^2 - x + 1 = 0\]One way to find the zeros is to try the Rational Root Theorem, which suggests possible rational roots. Trying $x = 1$:\[f(1) = 1^3 - 1^2 - 1 + 1 = 0\]Thus, $x = 1$ is a root. To find the other roots, perform synthetic or polynomial division using $x-1$. After dividing, we get:\[f(x) = (x-1)(x^2 + 0x - 1)\]Solve $x^2 - 1 = 0$ using the quadratic formula:\[x^2 - 1 = (x-1)(x+1)\]The other roots are $x = 1$ and $x = -1$. Hence, the zeros are $x = 1$ (multiplicity 2) and $x = -1$ (multiplicity 1).

Determine Graph Crossing or Touching Behavior

For each real zero, check the multiplicity to determine the graph's behavior:- At $x = 1$, the multiplicity is 2, meaning the graph will touch the x-axis and bounce back.- At $x = -1$, the multiplicity is 1, meaning the graph will cross the x-axis.

Finding the y-intercept and Points on the Graph

Find the y-intercept by evaluating $f(x)$ at $x = 0$:\[f(0) = 0^3 - 0^2 - 0 + 1 = 1\]Thus, the y-intercept is $(0, 1)$.To find additional points, choose values for $x$, calculate $f(x)$, and create points such as:$f(-2) = (-2)^3 - (-2)^2 - (-2) + 1 = -8 - 4 + 2 + 1 = -9$, so $(-2, -9)$.$f(2) = 2^3 - 2^2 - 2 + 1 = 8 - 4 - 2 + 1 = 3$, so $(2, 3)$.

Determine End Behavior

For the polynomial $f(x) = x^3 - x^2 - x + 1$, the degree is 3 (odd), and the leading coefficient is positive (1).- As $x \to -\infty$, $f(x) \to -\infty$.- As $x \to +\infty$, $f(x) \to +\infty$.

Sketch the Graph

To sketch the graph, plot the x-intercepts and observe the multiplicity (crossing or touching behavior), plot the y-intercept $(0,1)$, include a few additional points such as $(-2, -9)$ and $(2, 3)$. Add arrows to show end behavior:- The graph touches the x-axis at $x = 1$ with a bounce.- The graph crosses the x-axis at $x = -1$.- As $x$ moves towards infinity, the graph moves upwards, and as $x$ moves towards negative infinity, the graph moves downwards.

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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 $x$ where a polynomial function equals zero. In simpler terms, these are the points where the graph touches or crosses the x-axis. Consider the polynomial function $f(x) = x^3 - x^2 - x + 1$. To find the real zeros, you set the equation to zero and solve, which essentially means finding the roots of the equation.

Use the Rational Root Theorem to identify possible rational roots.
Test these roots to see if they satisfy the polynomial equation.
For this function, $x = 1$ is a real zero because substituting $x = 1$ into the polynomial results in zero, confirming it's a root.
Further solving gives us other zeros: $x = -1$ and $x = 1$ again making $x = 1$ a double root with multiplicity 2.

Understanding real zeros and their multiplicity helps in predicting the behavior of the graph at those points.

Graph Behavior

Understanding graph behavior is essential for interpreting how a polynomial function behaves at its zeros. At each zero:

If the zero has an odd multiplicity, the graph crosses the x-axis.
If the zero has an even multiplicity, the graph just touches and bounces off the x-axis without crossing it.

For our polynomial $f(x) = x^3 - x^2 - x + 1$:

At $x = 1$, with multiplicity 2, the graph touches and bounces at the x-axis. This results in a gentle curve at that point.
At $x = -1$, with multiplicity 1, the graph crosses the x-axis, indicating a straightforward transition from above the x-axis to below or vice versa.

This behavior is crucial in sketching accurate graphs for polynomial functions.

End Behavior

The end behavior of a polynomial function describes how the graph behaves as $x$ approaches positive or negative infinity. This is largely determined by the function's leading term. For the polynomial $f(x) = x^3 - x^2 - x + 1$, the leading term is $x^3$.

Since the degree of the polynomial (3) is odd, and the leading coefficient is positive, the end behavior follows a specific pattern.
As $x$ approaches positive infinity, $f(x)$ also approaches positive infinity.
Conversely, as $x$ approaches negative infinity, $f(x)$ approaches negative infinity.

This forms the basic guideline for predicting the shape and direction of the polynomial's graph at the extremes of the coordinate system.

Y-Intercept

The y-intercept is the point where a graph intersects the y-axis. For any function $f(x)$, this occurs when $x=0$. Calculating it involves evaluating the polynomial at zero.
For our polynomial $f(x) = x^3 - x^2 - x + 1$:

Plugging in zero, we find $f(0) = 0^3 - 0^2 - 0 + 1 = 1$.

This means the y-intercept is at the point $(0, 1)$. Knowing the y-intercept provides a starting reference on the graph and aids in its construction.
Additionally, calculating values of $f(x)$ at other points, like $f(2)$ or $f(-2)$, can help in understanding the shape of the graph further.

Rational Root Theorem

The Rational Root Theorem is a valuable tool in finding potential rational roots of a polynomial function. It states that if a polynomial has a rational root $\frac{p}{q}$, then $p$ (a factor of the constant term) and $q$ (a factor of the leading coefficient) are integers.

This theorem helps suggest possible candidates for real zeros.
For example, in $f(x) = x^3 - x^2 - x + 1$, possible roots might be the factors of 1 divided by the factors of 1, leading to potential roots like $ \pm 1$.
Testing these using substitution confirmed $x = 1$ as a root.

By narrowing down the possible rational roots, this theorem simplifies the often-complex process of solving polynomial equations.

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Short Answer

Step by step solution

Finding the Real Zeros

Determine Graph Crossing or Touching Behavior

Finding the y-intercept and Points on the Graph

Determine End Behavior

Sketch the Graph

Key Concepts

Real Zeros

Graph Behavior

End Behavior

Y-Intercept

Rational Root Theorem

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