Answer :
### Part D
Advantages of Writing the Polynomial Expression [tex]\(-7x^2 + 32x + 240\)[/tex] in Factored Form
1. Identification of Roots:
- When the polynomial is written in factored form, it becomes much easier to identify the roots of the equation. The roots are the values of [tex]\(x\)[/tex] where the polynomial equals zero. From our solution:
[tex]\[ x_1 = -4.0 \quad \text{and} \quad x_2 = 8.571428571428571 \][/tex]
2. Graphical Interpretation:
- The roots correspond to the points where the graph of the polynomial intersects the x-axis. Knowing these points aids in sketching the graph and understanding its behavior.
3. Behavior Analysis Near Roots:
- Factored form allows us to understand how the polynomial behaves as [tex]\(x\)[/tex] approaches the roots. For instance, in the interval between the roots, the polynomial either lies entirely above or below the x-axis depending on the leading coefficient and the nature of the roots.
4. Simplification for Calculus:
- If we need to perform calculus operations like differentiation or integration, a factored form can simplify these processes. For example, derivatives and integrals often have simpler forms when the function is factored.
5. Insights into Multiplicity of Roots:
- If any root repeats (has a multiplicity greater than one), this can be easily seen in the factored form. Multiplicity affects the shape of the graph at those roots, indicating where the graph touches or crosses the x-axis.
### Part E
Complete the Table for Different Values of [tex]\( x \)[/tex] in the Polynomial Expression [tex]\(-7x^2 + 32x + 240\)[/tex]:
Below is a table showing the [tex]\(y\)[/tex]-values for different [tex]\(x\)[/tex]-values in the polynomial expression:
[tex]\[ \begin{array}{| c | c |} \hline x & y = -7x^2 + 32x + 240 \\ \hline -10 & -780 \\ -9 & -615 \\ -8 & -464 \\ -7 & -327 \\ -6 & -204 \\ -5 & -95 \\ -4 & 0 \\ -3 & 81 \\ -2 & 148 \\ -1 & 201 \\ 0 & 240 \\ 1 & 265 \\ 2 & 276 \\ 3 & 273 \\ 4 & 256 \\ 5 & 225 \\ 6 & 180 \\ 7 & 121 \\ 8 & 48 \\ 9 & -39 \\ 10 & -140 \\ \hline \end{array} \][/tex]
Determining Key Features:
1. Vertex:
- The vertex of the parabola can provide insights into the maximum or minimum point of the polynomial.
2. Roots (Intercepts):
- From the table, we see that at [tex]\(x = -4\)[/tex] and [tex]\(x = 8.571428571428571\)[/tex], the polynomial crosses the x-axis, confirming the roots found earlier.
3. Behavior Analysis:
- [tex]\(y\)[/tex]-values switch signs between roots, giving us insights into where the polynomial crosses the x-axis.
4. Polynomial Behavior:
- For x values such as 0 and 1, we see the polynomial rises to a peak at [tex]\(x = 4\)[/tex] and then declines. This movement reflects the typical shape of a downward-opening parabola, indicating it has a maximum point.
In summary, writing a polynomial in factored form greatly aids in analysis by simplifying the identification of roots, enhancing graphical interpretation, and facilitating advanced operations such as differentiation and integration. The table of values demonstrates the polynomial's behavior, confirming the roots and showing the polynomial's parabolic nature.
Advantages of Writing the Polynomial Expression [tex]\(-7x^2 + 32x + 240\)[/tex] in Factored Form
1. Identification of Roots:
- When the polynomial is written in factored form, it becomes much easier to identify the roots of the equation. The roots are the values of [tex]\(x\)[/tex] where the polynomial equals zero. From our solution:
[tex]\[ x_1 = -4.0 \quad \text{and} \quad x_2 = 8.571428571428571 \][/tex]
2. Graphical Interpretation:
- The roots correspond to the points where the graph of the polynomial intersects the x-axis. Knowing these points aids in sketching the graph and understanding its behavior.
3. Behavior Analysis Near Roots:
- Factored form allows us to understand how the polynomial behaves as [tex]\(x\)[/tex] approaches the roots. For instance, in the interval between the roots, the polynomial either lies entirely above or below the x-axis depending on the leading coefficient and the nature of the roots.
4. Simplification for Calculus:
- If we need to perform calculus operations like differentiation or integration, a factored form can simplify these processes. For example, derivatives and integrals often have simpler forms when the function is factored.
5. Insights into Multiplicity of Roots:
- If any root repeats (has a multiplicity greater than one), this can be easily seen in the factored form. Multiplicity affects the shape of the graph at those roots, indicating where the graph touches or crosses the x-axis.
### Part E
Complete the Table for Different Values of [tex]\( x \)[/tex] in the Polynomial Expression [tex]\(-7x^2 + 32x + 240\)[/tex]:
Below is a table showing the [tex]\(y\)[/tex]-values for different [tex]\(x\)[/tex]-values in the polynomial expression:
[tex]\[ \begin{array}{| c | c |} \hline x & y = -7x^2 + 32x + 240 \\ \hline -10 & -780 \\ -9 & -615 \\ -8 & -464 \\ -7 & -327 \\ -6 & -204 \\ -5 & -95 \\ -4 & 0 \\ -3 & 81 \\ -2 & 148 \\ -1 & 201 \\ 0 & 240 \\ 1 & 265 \\ 2 & 276 \\ 3 & 273 \\ 4 & 256 \\ 5 & 225 \\ 6 & 180 \\ 7 & 121 \\ 8 & 48 \\ 9 & -39 \\ 10 & -140 \\ \hline \end{array} \][/tex]
Determining Key Features:
1. Vertex:
- The vertex of the parabola can provide insights into the maximum or minimum point of the polynomial.
2. Roots (Intercepts):
- From the table, we see that at [tex]\(x = -4\)[/tex] and [tex]\(x = 8.571428571428571\)[/tex], the polynomial crosses the x-axis, confirming the roots found earlier.
3. Behavior Analysis:
- [tex]\(y\)[/tex]-values switch signs between roots, giving us insights into where the polynomial crosses the x-axis.
4. Polynomial Behavior:
- For x values such as 0 and 1, we see the polynomial rises to a peak at [tex]\(x = 4\)[/tex] and then declines. This movement reflects the typical shape of a downward-opening parabola, indicating it has a maximum point.
In summary, writing a polynomial in factored form greatly aids in analysis by simplifying the identification of roots, enhancing graphical interpretation, and facilitating advanced operations such as differentiation and integration. The table of values demonstrates the polynomial's behavior, confirming the roots and showing the polynomial's parabolic nature.
