Question
create a function of degree 3 with the roots, c = 5, 0, -2. Then create a function with the same roots but with degree 6.
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Answer to a math question create a function of degree 3 with the roots, c = 5, 0, -2. Then create a function with the same roots but with degree 6.
Gerhard
4.592 Answers
1. Given the polynomial of degree 3 with roots \( 5, 0, -2 \):
f(x) = k(x - 5)x(x + 2)
Expanding it:
f(x) = k(x^3 - 3x^2 - 10x)
2. For the polynomial of degree 6 with the same roots but each root appearing twice:
g(x) = m(x - 5)^2 x^2 (x + 2)^2
Expanding this expression step-by-step:
g(x) = m(x^2 - 10x + 25)x^2(x^2 + 4x + 4)
First, expand the product of the quadratic terms:
(x^2 - 10x + 25)(x^2 + 4x + 4) = x^4 + 4x^3 + 4x^2 - 10x^3 - 40x^2 - 40x + 25x^2 + 100x + 100
Combine like terms:
= x^4 + (4x^3 - 10x^3) + (4x^2 - 40x^2 + 25x^2) + (-40x + 100x) + 100
= x^4 - 6x^3 - 11x^2 + 60x + 100
Then multiply by \( x^2 \):
g(x) = m x^2 (x^4 - 6x^3 - 11x^2 + 60x + 100)
= m (x^6 - 6x^5 - 11x^4 + 60x^3 + 100x^2)
Expanding it:
2. For the polynomial of degree 6 with the same roots but each root appearing twice:
Expanding this expression step-by-step:
First, expand the product of the quadratic terms:
Combine like terms:
Then multiply by \( x^2 \):
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