Problem 57
Question
This set of exercises will draw on the ideas presented in this section and your general math background. Find a polynomial function whose zeros are \(x=0,1\) and \(-1 .\) Is your answer the only correct answer? Why or why not? You may confirm your answer with a graphing utility.
Step-by-Step Solution
Verified Answer
A polynomial function with zeros at 0, 1, and -1 can be \(f(x) = a \cdot x \cdot (x - 1) \cdot (x + 1)\) or \(f(x) = a \cdot (x^3 - x)\). The answer is not unique: for any non-zero \(a\), we will get a function with these zeros. Verified with a graphing utility.
1Step 1 - Formulate the polynomial function from the given zeros
The zeros are \(x = 0, 1, -1\). Therefore, a polynomial function with these zeros can be formulated as \(f(x) = a \cdot (x - 0) \cdot (x - 1) \cdot (x - (-1))\), which simplifies to \(f(x) = a \cdot x \cdot (x - 1) \cdot (x + 1)\). 'a' is any non-zero real number called the leading coefficient.
2Step 2 - Expand the equation
We can keep our polynomial in factored form as \(f(x) = a \cdot x \cdot (x - 1) \cdot (x + 1)\), or alternatively expand it to \(f(x) = a \cdot (x^3 - x)\).
3Step 3 - Address the uniqueness of the solution
The solution is not unique. The form of the polynomial \(f(x) = a \cdot x \cdot (x - 1) \cdot (x + 1)\) or \(f(x) = a \cdot (x^3 - x)\) depends on the arbitrary choice of the leading coefficient \(a\). For different values of \(a\), we can have infinitely many different polynomials, all with zeros at \(0, 1, -1\). any non-zero value of \(a\) is a possible leading coefficient.
4Step 4 - Confirm solution with a graphing utility
You can use a graphing utility to plot the function and confirm that the zeros of the polynomial are indeed at \(x = 0, 1, -1\). This exercise does not require a specific value for \(a\), the effect of changing the leading coefficient can also be observed.
Key Concepts
Factoring PolynomialsLeading CoefficientGraphing Polynomials
Factoring Polynomials
Understanding how to factor polynomials is essential for solving various types of algebraic problems. In the context of finding zeros of a polynomial function, like in our exercise, factoring is the process of breaking down the polynomial into simpler expressions, which represent the solutions or roots where the polynomial equals zero.
Imagine you have a large product of numbers, and you know that the product equals zero. If any single number in that product is zero, the entire product becomes zero. That's the principle used when factoring polynomials to find their zeros.
The given exercise involves a polynomial that has been factored into \(x\), \(x - 1\), and \(x + 1\), each corresponding to a zero of the polynomial. By setting each factor equal to zero, you can solve for the values of \(x\) which make the function equal to zero, hence finding the zeros: \(x = 0\), \(x = 1\), and \(x = -1\). These are the exact values given in the original problem!
Imagine you have a large product of numbers, and you know that the product equals zero. If any single number in that product is zero, the entire product becomes zero. That's the principle used when factoring polynomials to find their zeros.
The given exercise involves a polynomial that has been factored into \(x\), \(x - 1\), and \(x + 1\), each corresponding to a zero of the polynomial. By setting each factor equal to zero, you can solve for the values of \(x\) which make the function equal to zero, hence finding the zeros: \(x = 0\), \(x = 1\), and \(x = -1\). These are the exact values given in the original problem!
Leading Coefficient
The leading coefficient of a polynomial is significant as it influences the end behavior of the polynomial's graph. It is the coefficient of the term with the highest power of \(x\) once the polynomial is in its standard form. For a given set of zeros, the leading coefficient, usually denoted as \(a\), can be any non-zero value. It affects the steepness and direction of the graph but not the actual zeros of the polynomial.
In our exercise, we denoted the leading coefficient by \(a\) in the polynomial function \(f(x) = a \cdot x \cdot (x - 1) \cdot (x + 1)\), which correlates to the simplest form \(f(x) = a \cdot (x^3 - x)\). The value of \(a\) dictates whether the graph opens upwards or downwards and how quickly it increases or decreases as \(x\) moves away from the zeros. It is important to note that when \(a\) is positive, the graph will generally open upwards, and when \(a\) is negative, the graph will open downwards. As such, the leading coefficient plays a crucial role in the shape of the polynomial's graph even though it does not change the roots of the polynomial itself.
In our exercise, we denoted the leading coefficient by \(a\) in the polynomial function \(f(x) = a \cdot x \cdot (x - 1) \cdot (x + 1)\), which correlates to the simplest form \(f(x) = a \cdot (x^3 - x)\). The value of \(a\) dictates whether the graph opens upwards or downwards and how quickly it increases or decreases as \(x\) moves away from the zeros. It is important to note that when \(a\) is positive, the graph will generally open upwards, and when \(a\) is negative, the graph will open downwards. As such, the leading coefficient plays a crucial role in the shape of the polynomial's graph even though it does not change the roots of the polynomial itself.
Graphing Polynomials
Graphing polynomials is a valuable skill for visualizing the behavior of polynomial functions and verifying their properties. When we graph polynomials, we are essentially plotting the curve that represents all the possible points \( (x, y) \) that satisfy the polynomial equation \( f(x) \).
The zeros of the polynomial, which are the solutions we discussed in factoring, are the points where the graph intersects the x-axis. These intersections correspond to the real roots of the polynomial. In the original exercise, using a graphing utility can confirm the zeros at \(x = 0\), \(x = 1\), and \(x = -1\), where the graph touches or crosses the x-axis.
Additionally, the leading coefficient, as mentioned previously, will affect the way the graph behaves, especially at the ends of the graph, known as the end behavior. By examining a graph, one can not only ensure the correctness of the zeros but also understand how alterations in the leading coefficient \(a\) affect the steepness and direction of the graph. Plotting a polynomial can offer significant insight into the overall shape and characteristics of the function beyond just the zeros.
The zeros of the polynomial, which are the solutions we discussed in factoring, are the points where the graph intersects the x-axis. These intersections correspond to the real roots of the polynomial. In the original exercise, using a graphing utility can confirm the zeros at \(x = 0\), \(x = 1\), and \(x = -1\), where the graph touches or crosses the x-axis.
Additionally, the leading coefficient, as mentioned previously, will affect the way the graph behaves, especially at the ends of the graph, known as the end behavior. By examining a graph, one can not only ensure the correctness of the zeros but also understand how alterations in the leading coefficient \(a\) affect the steepness and direction of the graph. Plotting a polynomial can offer significant insight into the overall shape and characteristics of the function beyond just the zeros.
Other exercises in this chapter
Problem 57
Graph the function using a graphing utility, and find its zeros. $$p(x)=-2 x^{4}+13 x^{3}-23 x^{2}+3 x+9$$
View solution Problem 57
For each polynomial function, (a) find a function of the form \(y=c x^{2}\) that has the same end behavior. (b) find the \(x\) - and \(y\) -intercept(s) of the
View solution Problem 58
Sketch a graph of the rational function and find all intercepts and slant asymptotes. Indicate all asymptotes onthe graph. $$h(x)=\frac{x^{2}-9}{x}$$
View solution Problem 58
Graph the function using a graphing utility, and find its zeros. $$f(x)=x^{3}+x^{2}+x-3.1 x^{2}-2.5 x-4$$
View solution