Westonci.ca connects you with experts who provide insightful answers to your questions. Join us today and start learning! Connect with a community of experts ready to provide precise solutions to your questions on our user-friendly Q&A platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
Let's tackle the task step-by-step based on the initial steps you provided for Hill 1 and Hill 2.
### Hill 1:
You want a polynomial with three peaks, and you have chosen three points on the [tex]\( x \)[/tex]-axis (the zeros), which are [tex]\( x = 1 \)[/tex], [tex]\( x = 3 \)[/tex], and [tex]\( x = 4 \)[/tex]. Representing each zero as a factor, you built the polynomial:
[tex]\[ F(x) = (x - 1)(x - 3)(x - 4) \][/tex]
Now, let's expand this polynomial:
1. First, expand [tex]\( (x - 1)(x - 3) \)[/tex]:
[tex]\[ (x - 1)(x - 3) = x^2 - 3x - x + 3 = x^2 - 4x + 3 \][/tex]
2. Next, expand this result with [tex]\( (x - 4) \)[/tex]:
[tex]\[(x^2 - 4x + 3)(x - 4)\][/tex]
Distribute each term in [tex]\( x^2 - 4x + 3 \)[/tex] by [tex]\( x - 4 \)[/tex]:
- [tex]\( x^2(x - 4) = x^3 - 4x^2 \)[/tex]
- [tex]\( -4x(x - 4) = -4x^2 + 16x \)[/tex]
- [tex]\( 3(x - 4) = 3x - 12 \)[/tex]
Combine these results:
[tex]\[ x^3 - 4x^2 - 4x^2 + 16x + 3x - 12 = x^3 - 8x^2 + 19x - 12 \][/tex]
### Hill 2:
Your second layer should be a transformation of the first polynomial, [tex]\( F(x) \)[/tex]. You indicated that this transformation involves multiplying the polynomial by [tex]\(-12\)[/tex] or by some combination involving [tex]\(-12\)[/tex].
We can suppose you meant multiplying [tex]\( F(x) \)[/tex] by [tex]\(-12 \)[/tex]:
[tex]\[ G(x) = -12 \cdot (x^3 - 8x^2 + 19x - 12) \][/tex]
Now, let's distribute [tex]\(-12\)[/tex] throughout the polynomial:
[tex]\[ G(x) = -12x^3 + 96x^2 - 228x + 144 \][/tex]
### Summary:
- Hill 1: The polynomial [tex]\( F(x) = (x-1)(x-3)(x-4) \)[/tex] gives [tex]\( F(x) = x^3 - 8x^2 + 19x - 12 \)[/tex].
- Hill 2: Transforming [tex]\( F(x) \)[/tex] by multiplying it by [tex]\(-12\)[/tex] results in [tex]\( G(x) = -12x^3 + 96x^2 - 228x + 144 \)[/tex].
These are the detailed steps to create and transform the polynomials for Hill 1 and Hill 2.
### Hill 1:
You want a polynomial with three peaks, and you have chosen three points on the [tex]\( x \)[/tex]-axis (the zeros), which are [tex]\( x = 1 \)[/tex], [tex]\( x = 3 \)[/tex], and [tex]\( x = 4 \)[/tex]. Representing each zero as a factor, you built the polynomial:
[tex]\[ F(x) = (x - 1)(x - 3)(x - 4) \][/tex]
Now, let's expand this polynomial:
1. First, expand [tex]\( (x - 1)(x - 3) \)[/tex]:
[tex]\[ (x - 1)(x - 3) = x^2 - 3x - x + 3 = x^2 - 4x + 3 \][/tex]
2. Next, expand this result with [tex]\( (x - 4) \)[/tex]:
[tex]\[(x^2 - 4x + 3)(x - 4)\][/tex]
Distribute each term in [tex]\( x^2 - 4x + 3 \)[/tex] by [tex]\( x - 4 \)[/tex]:
- [tex]\( x^2(x - 4) = x^3 - 4x^2 \)[/tex]
- [tex]\( -4x(x - 4) = -4x^2 + 16x \)[/tex]
- [tex]\( 3(x - 4) = 3x - 12 \)[/tex]
Combine these results:
[tex]\[ x^3 - 4x^2 - 4x^2 + 16x + 3x - 12 = x^3 - 8x^2 + 19x - 12 \][/tex]
### Hill 2:
Your second layer should be a transformation of the first polynomial, [tex]\( F(x) \)[/tex]. You indicated that this transformation involves multiplying the polynomial by [tex]\(-12\)[/tex] or by some combination involving [tex]\(-12\)[/tex].
We can suppose you meant multiplying [tex]\( F(x) \)[/tex] by [tex]\(-12 \)[/tex]:
[tex]\[ G(x) = -12 \cdot (x^3 - 8x^2 + 19x - 12) \][/tex]
Now, let's distribute [tex]\(-12\)[/tex] throughout the polynomial:
[tex]\[ G(x) = -12x^3 + 96x^2 - 228x + 144 \][/tex]
### Summary:
- Hill 1: The polynomial [tex]\( F(x) = (x-1)(x-3)(x-4) \)[/tex] gives [tex]\( F(x) = x^3 - 8x^2 + 19x - 12 \)[/tex].
- Hill 2: Transforming [tex]\( F(x) \)[/tex] by multiplying it by [tex]\(-12\)[/tex] results in [tex]\( G(x) = -12x^3 + 96x^2 - 228x + 144 \)[/tex].
These are the detailed steps to create and transform the polynomials for Hill 1 and Hill 2.
We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.