Form a polynomial function whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient. Zeros: -1, multiplicity 1, 3, multiplicity 2, degree 3

Reggie 2021-02-21 Answered
Form a polynomial function whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient. Zeros: -1, multiplicity 1, 3, multiplicity 2, degree 3
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Fatema Sutton
Answered 2021-02-22 Author has 88 answers
f(x)=(x+1)(x3)2=x35x2+3x+9
f(x)=x35x35x2=3x+9
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This seems very obvious and I am having a bit of trouble producing a formal proof.
sketch proof that the composition of two polynomials is a polynomial
Let
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sketch proof that the composition of two rational functions is a rational function
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Let
a ( z 1 ) = p ( z 1 ) q ( z 1 ) ,   b ( z 2 ) = p ( z 2 ) q ( z 2 )
Now,
( a b ) ( z 2 ) = a ( b ( z 2 ) )           (by definition) = p ( p ( z 2 ) q ( z 2 ) ) q ( p ( z 2 ) q ( z 2 ) ) = a n ( p ( z 2 ) q ( z 2 ) ) n + a n 1 ( p ( z 2 ) q ( z 2 ) ) n 1 + . . . + a 1 ( p ( z 2 ) q ( z 2 ) ) + a 0 b n ( p ( z 2 ) q ( z 2 ) ) n + b n 1 ( p ( z 2 ) q ( z 2 ) ) n 1 + . . . + b 1 ( p ( z 2 ) q ( z 2 ) ) + b 0
Notice that ( p ( z 2 ) q ( z 2 ) ) i         ( i = n , n 1 , . . , 0 ) is a polynomial as
( f g ) ( z 2 ) = f ( g ( z 2 ) ) = ( p ( z 2 ) q ( z 2 ) ) i
where
f ( x ) = x i ,     g ( z 2 ) = ( p ( z 2 ) q ( z 2 ) )
are both polynomials. Hence ( a b ) ( z 2 ) is a rational function as it is the quotient of polynomials.
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