Problem 59
Question
The equations have real roots that are rational. Use the Rational Zero Theorem to list all possible rational roots. Then graph the polynomial function in the given viewing rectangle to determine which possible rational roots are actual roots of the equation. $$ 4 x^{4}+4 x^{3}+7 x^{2}-x-2=0 ;[-2,2,1] \text { by }[-5,5,1] $$
Step-by-Step Solution
Verified Answer
The rational roots of the equation could be any among ±1, ±1/2, ±1/4, ±2 but to verify which one of these are the actual roots, you'll have to plot the graph and observe where it crosses the x-axis. Each intersection point reveals a real root of the equation.
1Step 1: Find the Possible Rational Roots
To find all possible rational roots, we will take the factors of the constant term and divide them by the factors of the leading coefficient. The constant term in this equation is -2 and its factors are: -1, 1, -2, 2. The leading coefficient is 4, and its factors are -1, 1, -2, 2, -4, 4. Thus, all possible rational roots for this equation are: ±1/1, ±1/2, ±1/4, ±2/1, ±2/2, ±2/4, simplifying, we get: ±1, ±1/2, ±1/4, ±2.
2Step 2: Graph the Equation
Now, plot the equation on a graph within the viewing rectangle [-2,2,1] by [-5,5,1] and see which of the possible rational roots we found are actual zeros of the equation. The roots of the equation correspond to the x-values where the graph crosses the x-axis.
3Step 3: Determine the actual roots
By observing the generated graph, we can see where the graph crosses the x-axis. These intersections are the real roots of the polynomial. Those of our possible rational roots that match these intersections can be deemed the actual rational roots of our equation.
Key Concepts
Polynomial FunctionRational RootsGraphing Polynomials
Polynomial Function
A polynomial function is an expression made up of variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer exponentiation of variables. An example of a polynomial function is the equation \(4x^4+4x^3+7x^2-x-2=0\), given in the exercise.
When dealing with polynomial functions, one aspect that often needs to be determined is its roots, which are the values of \(x\) that make the function equal to zero. Finding the roots is vital as it helps in understanding the behavior of the function. For a polynomial of degree \(n\), there can be up to \(n\) real roots.
Polynomial functions can be graphed on a coordinate plane to show their overall shape, turning points, and places where the graph crosses the x-axis (its zeros). Graphing them especially within a viewing rectangle as the problem suggests, \( [-2,2,1] \text{ by } [-5,5,1]\), helps us visually confirm where the polynomial might have its roots.
When dealing with polynomial functions, one aspect that often needs to be determined is its roots, which are the values of \(x\) that make the function equal to zero. Finding the roots is vital as it helps in understanding the behavior of the function. For a polynomial of degree \(n\), there can be up to \(n\) real roots.
Polynomial functions can be graphed on a coordinate plane to show their overall shape, turning points, and places where the graph crosses the x-axis (its zeros). Graphing them especially within a viewing rectangle as the problem suggests, \( [-2,2,1] \text{ by } [-5,5,1]\), helps us visually confirm where the polynomial might have its roots.
Rational Roots
Rational roots of a polynomial function are any roots that can be expressed as a fraction \(\frac{p}{q}\), where \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient. The Rational Zero Theorem provides us with a systematic way to list all possible rational roots of a polynomial equation.
In the exercise, the rational zero theorem is applied by taking the factors of the constant term (-2) and dividing them by the factors of the leading coefficient (4). This renders a list of possible rational roots: \(\pm 1\), \(\pm \frac{1}{2}\), \(\pm \frac{1}{4}\), and \(\pm 2\). Not every potential root listed by the theorem is an actual root, which is why testing them or graphing the polynomial (as in Step 2 of the solution) is important to confirm which of these rational roots are actual roots.
In the exercise, the rational zero theorem is applied by taking the factors of the constant term (-2) and dividing them by the factors of the leading coefficient (4). This renders a list of possible rational roots: \(\pm 1\), \(\pm \frac{1}{2}\), \(\pm \frac{1}{4}\), and \(\pm 2\). Not every potential root listed by the theorem is an actual root, which is why testing them or graphing the polynomial (as in Step 2 of the solution) is important to confirm which of these rational roots are actual roots.
Graphing Polynomials
Graphing polynomials is a practical method for visualizing the shape and finding the roots of the function. By plotting a polynomial function on a coordinate system, you are able to see the curve it forms which can provide insights into its behavior such as intercepts, turning points, and the intervals where the function is positive or negative.
The process of graphing involves creating a table of values by substituting various x-values into the polynomial and calculating the corresponding y-values. The points are then marked and connected to display the function's curve. The viewing rectangle in the problem, \( [-2,2,1] \text{ by } [-5,5,1]\), tells us the scale for the x-axis and the y-axis respectively and helps us to focus on an appropriate section of the graph. This is particularly useful when attempting to locate the x-intercepts, which give us the actual roots of the polynomial function as demonstrated in Step 2 of the problem's solution.
The process of graphing involves creating a table of values by substituting various x-values into the polynomial and calculating the corresponding y-values. The points are then marked and connected to display the function's curve. The viewing rectangle in the problem, \( [-2,2,1] \text{ by } [-5,5,1]\), tells us the scale for the x-axis and the y-axis respectively and helps us to focus on an appropriate section of the graph. This is particularly useful when attempting to locate the x-intercepts, which give us the actual roots of the polynomial function as demonstrated in Step 2 of the problem's solution.
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