Answer :
Sure! Let's address each part of the question step-by-step:
1. Graph of the Polynomial Function:
For the polynomial [tex]\( f(x) = x^4 + x^3 - 2x^2 \)[/tex], to determine where the graph crosses or touches the x-axis, we need to factor and analyze the roots.
- First, factor out the common term:
[tex]\( f(x) = x^2(x^2 + x - 2) \)[/tex]
- Factor the quadratic:
[tex]\( x^2 + x - 2 = (x - 1)(x + 2) \)[/tex]
- So, the function becomes:
[tex]\( f(x) = x^2(x - 1)(x + 2) \)[/tex]
The roots are [tex]\( x = 0, x = 1, \)[/tex] and [tex]\( x = -2 \)[/tex].
- At [tex]\( x = 0 \)[/tex], the factor [tex]\( x^2 \)[/tex] indicates that the graph touches the x-axis (since it is a repeated root).
- At [tex]\( x = 1 \)[/tex] and [tex]\( x = -2 \)[/tex], the graph crosses the x-axis (because these roots have odd multiplicities).
Therefore, the correct statement that describes the graph is:
- c. The graph crosses the [tex]\( x \)[/tex]-axis at [tex]\( x = -2 \)[/tex] and [tex]\( x = 1 \)[/tex] and touches the [tex]\( x \)[/tex]-axis at [tex]\( x = 0 \)[/tex].
2. Roots of a Third Degree Polynomial:
If one root is [tex]\( -5 + 2i \)[/tex], then its conjugate, [tex]\( -5 - 2i \)[/tex], must also be a root (because polynomial coefficients are real). For a third-degree polynomial, we need one more root, which can be real or complex (without imaginary part). So, there are three roots in total: [tex]\( -5 + 2i \)[/tex], [tex]\( -5 - 2i \)[/tex], and one real root.
3. Conjugate Root Theorem:
If [tex]\( 5 + 6i \)[/tex] is a root, then its conjugate, [tex]\( 5 - 6i \)[/tex], must also be a root.
4. Degree of the Polynomial:
Sally knows three roots: [tex]\( -5, 3i, \)[/tex] and [tex]\( 5 \)[/tex]. The root [tex]\( 3i \)[/tex] implies its conjugate [tex]\( -3i \)[/tex] is also a root. So, there are actually four roots: [tex]\( -5, 5, 3i, \)[/tex] and [tex]\( -3i \)[/tex]. Thus, the polynomial must have a degree of at least 4. Therefore, Sally's conclusion is incorrect.
5. Roots of the Polynomial Function:
[tex]\( f(x) = (x-2)^4(x+5)^3 \)[/tex]
- Roots are [tex]\( x = 2 \)[/tex] and [tex]\( x = -5 \)[/tex].
- [tex]\( x = 2 \)[/tex] has a multiplicity of 4, so the graph touches the x-axis at this point but does not cross it.
- [tex]\( x = -5 \)[/tex] has a multiplicity of 3, so the graph crosses the x-axis at this point.
6. Fundamental Theorem of Algebra:
For the polynomial [tex]\( f(x) = 8x^9 - x^5 + x^3 + 6 \)[/tex], the degree is 9. Therefore, it has 9 roots in the complex number system.
7. X-Intercepts of the Polynomial:
Hypothetical polynomial given [tex]\( f(x) = x^4 - 7x^2 \)[/tex] can be rewritten as:
[tex]\( f(x) = x^2(x^2 - 7) \)[/tex]
- The x-intercepts occur at the zeroes of the polynomial: [tex]\( x = 0 \)[/tex] and [tex]\( x = \sqrt{7}, -\sqrt{7} \)[/tex].
- So the graph has 3 x-intercepts.
8. Remaining Roots:
Given [tex]\( f(x) = 60x^4 + 86x^3 - 46x^2 - 43x + 8 \)[/tex] with roots [tex]\( \frac{8}{5} \)[/tex] and [tex]\( \frac{1}{0} \)[/tex], since [tex]\(\frac{1}{0}\)[/tex] is not a valid root (undefined division), review the prompt context for further information, and ensure that another method is used to determine the roots.
9. Write a Quadratic Equation with Given Roots:
Not provided, so if you're tasked with creating a quadratic polynomial from roots [tex]\( a \)[/tex] and [tex]\( b \)[/tex], it can be expressed as [tex]\( (x-a)(x-b) = 0 \)[/tex] or expanded [tex]\( x^2 - (a+b)x + ab = 0 \)[/tex].
I hope this helps clarify the steps for analyzing and describing polynomial functions! If you have any more questions, don't hesitate to ask.
1. Graph of the Polynomial Function:
For the polynomial [tex]\( f(x) = x^4 + x^3 - 2x^2 \)[/tex], to determine where the graph crosses or touches the x-axis, we need to factor and analyze the roots.
- First, factor out the common term:
[tex]\( f(x) = x^2(x^2 + x - 2) \)[/tex]
- Factor the quadratic:
[tex]\( x^2 + x - 2 = (x - 1)(x + 2) \)[/tex]
- So, the function becomes:
[tex]\( f(x) = x^2(x - 1)(x + 2) \)[/tex]
The roots are [tex]\( x = 0, x = 1, \)[/tex] and [tex]\( x = -2 \)[/tex].
- At [tex]\( x = 0 \)[/tex], the factor [tex]\( x^2 \)[/tex] indicates that the graph touches the x-axis (since it is a repeated root).
- At [tex]\( x = 1 \)[/tex] and [tex]\( x = -2 \)[/tex], the graph crosses the x-axis (because these roots have odd multiplicities).
Therefore, the correct statement that describes the graph is:
- c. The graph crosses the [tex]\( x \)[/tex]-axis at [tex]\( x = -2 \)[/tex] and [tex]\( x = 1 \)[/tex] and touches the [tex]\( x \)[/tex]-axis at [tex]\( x = 0 \)[/tex].
2. Roots of a Third Degree Polynomial:
If one root is [tex]\( -5 + 2i \)[/tex], then its conjugate, [tex]\( -5 - 2i \)[/tex], must also be a root (because polynomial coefficients are real). For a third-degree polynomial, we need one more root, which can be real or complex (without imaginary part). So, there are three roots in total: [tex]\( -5 + 2i \)[/tex], [tex]\( -5 - 2i \)[/tex], and one real root.
3. Conjugate Root Theorem:
If [tex]\( 5 + 6i \)[/tex] is a root, then its conjugate, [tex]\( 5 - 6i \)[/tex], must also be a root.
4. Degree of the Polynomial:
Sally knows three roots: [tex]\( -5, 3i, \)[/tex] and [tex]\( 5 \)[/tex]. The root [tex]\( 3i \)[/tex] implies its conjugate [tex]\( -3i \)[/tex] is also a root. So, there are actually four roots: [tex]\( -5, 5, 3i, \)[/tex] and [tex]\( -3i \)[/tex]. Thus, the polynomial must have a degree of at least 4. Therefore, Sally's conclusion is incorrect.
5. Roots of the Polynomial Function:
[tex]\( f(x) = (x-2)^4(x+5)^3 \)[/tex]
- Roots are [tex]\( x = 2 \)[/tex] and [tex]\( x = -5 \)[/tex].
- [tex]\( x = 2 \)[/tex] has a multiplicity of 4, so the graph touches the x-axis at this point but does not cross it.
- [tex]\( x = -5 \)[/tex] has a multiplicity of 3, so the graph crosses the x-axis at this point.
6. Fundamental Theorem of Algebra:
For the polynomial [tex]\( f(x) = 8x^9 - x^5 + x^3 + 6 \)[/tex], the degree is 9. Therefore, it has 9 roots in the complex number system.
7. X-Intercepts of the Polynomial:
Hypothetical polynomial given [tex]\( f(x) = x^4 - 7x^2 \)[/tex] can be rewritten as:
[tex]\( f(x) = x^2(x^2 - 7) \)[/tex]
- The x-intercepts occur at the zeroes of the polynomial: [tex]\( x = 0 \)[/tex] and [tex]\( x = \sqrt{7}, -\sqrt{7} \)[/tex].
- So the graph has 3 x-intercepts.
8. Remaining Roots:
Given [tex]\( f(x) = 60x^4 + 86x^3 - 46x^2 - 43x + 8 \)[/tex] with roots [tex]\( \frac{8}{5} \)[/tex] and [tex]\( \frac{1}{0} \)[/tex], since [tex]\(\frac{1}{0}\)[/tex] is not a valid root (undefined division), review the prompt context for further information, and ensure that another method is used to determine the roots.
9. Write a Quadratic Equation with Given Roots:
Not provided, so if you're tasked with creating a quadratic polynomial from roots [tex]\( a \)[/tex] and [tex]\( b \)[/tex], it can be expressed as [tex]\( (x-a)(x-b) = 0 \)[/tex] or expanded [tex]\( x^2 - (a+b)x + ab = 0 \)[/tex].
I hope this helps clarify the steps for analyzing and describing polynomial functions! If you have any more questions, don't hesitate to ask.
