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Answer :
To solve this problem, we'll go through each part step by step.
### Part (a): Sketching the Graph
The given quartic equation is:
[tex]\[ y = x^4 + 3x^3 - 35x^2 - 39x + 70 \][/tex]
Steps to Sketch the Graph:
1. Determine the x-intercepts: These occur where [tex]\( y = 0 \)[/tex]. From the calculations, the x-intercepts are at [tex]\( x = -7, -2, 1, \)[/tex] and [tex]\( 5 \)[/tex]. These points are where the graph crosses the x-axis.
2. Identify the behavior at the x-intercepts:
- Since these intercepts are roots of a polynomial, at each root the graph crosses the x-axis.
3. Plot the x-intercepts on the graph:
- Mark points on the x-axis at [tex]\( x = -7, -2, 1, \)[/tex] and [tex]\( 5 \)[/tex].
4. Consider the end behavior:
- For a quartic equation, if the leading coefficient (coefficient of [tex]\( x^4 \)[/tex]) is positive, the ends of the graph will rise on both sides.
5. Sketch the curve smoothly connecting these intercepts and consistent with the end behavior.
### Part (b): Factored Form
The quartic equation can be expressed in its factored form based on the x-intercepts. Given the roots [tex]\( x = -7, -2, 1, \)[/tex] and [tex]\( 5 \)[/tex], the factored form of the polynomial is:
[tex]\[ (x - 5)(x - 1)(x + 2)(x + 7) \][/tex]
Here's how we derive this:
- Each x-intercept corresponds to a factor of the equation.
- To form the factors, subtract each root from [tex]\( x \)[/tex]:
- For [tex]\( x = 5 \)[/tex], the factor is [tex]\( (x - 5) \)[/tex].
- For [tex]\( x = 1 \)[/tex], the factor is [tex]\( (x - 1) \)[/tex].
- For [tex]\( x = -2 \)[/tex], the factor is [tex]\( (x + 2) \)[/tex].
- For [tex]\( x = -7 \)[/tex], the factor is [tex]\( (x + 7) \)[/tex].
Thus, the expression is represented in its factored form:
[tex]\[ (x - 5)(x - 1)(x + 2)(x + 7) \][/tex]
This completes the solution to the problem. You have correctly identified the x-intercepts and expressed the polynomial in its factored form. Just remember to sketch your graph using these steps and you will have a successful representation of the quartic function.
### Part (a): Sketching the Graph
The given quartic equation is:
[tex]\[ y = x^4 + 3x^3 - 35x^2 - 39x + 70 \][/tex]
Steps to Sketch the Graph:
1. Determine the x-intercepts: These occur where [tex]\( y = 0 \)[/tex]. From the calculations, the x-intercepts are at [tex]\( x = -7, -2, 1, \)[/tex] and [tex]\( 5 \)[/tex]. These points are where the graph crosses the x-axis.
2. Identify the behavior at the x-intercepts:
- Since these intercepts are roots of a polynomial, at each root the graph crosses the x-axis.
3. Plot the x-intercepts on the graph:
- Mark points on the x-axis at [tex]\( x = -7, -2, 1, \)[/tex] and [tex]\( 5 \)[/tex].
4. Consider the end behavior:
- For a quartic equation, if the leading coefficient (coefficient of [tex]\( x^4 \)[/tex]) is positive, the ends of the graph will rise on both sides.
5. Sketch the curve smoothly connecting these intercepts and consistent with the end behavior.
### Part (b): Factored Form
The quartic equation can be expressed in its factored form based on the x-intercepts. Given the roots [tex]\( x = -7, -2, 1, \)[/tex] and [tex]\( 5 \)[/tex], the factored form of the polynomial is:
[tex]\[ (x - 5)(x - 1)(x + 2)(x + 7) \][/tex]
Here's how we derive this:
- Each x-intercept corresponds to a factor of the equation.
- To form the factors, subtract each root from [tex]\( x \)[/tex]:
- For [tex]\( x = 5 \)[/tex], the factor is [tex]\( (x - 5) \)[/tex].
- For [tex]\( x = 1 \)[/tex], the factor is [tex]\( (x - 1) \)[/tex].
- For [tex]\( x = -2 \)[/tex], the factor is [tex]\( (x + 2) \)[/tex].
- For [tex]\( x = -7 \)[/tex], the factor is [tex]\( (x + 7) \)[/tex].
Thus, the expression is represented in its factored form:
[tex]\[ (x - 5)(x - 1)(x + 2)(x + 7) \][/tex]
This completes the solution to the problem. You have correctly identified the x-intercepts and expressed the polynomial in its factored form. Just remember to sketch your graph using these steps and you will have a successful representation of the quartic function.
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