Find a polynomial of degree 3 having with the zeros 1+sqrt(2),1-sqrt(2), and 4.

sengihantq 2022-10-08 Answered
Find a polynomial of degree 3 having with the zeros 1 + 2 , 1 2 , and 4.
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Answers (1)

selixaak
Answered 2022-10-09 Author has 6 answers
If a is a zero of polynomial, then (x-a) is its factor.
Given three zeros are 1 + 2 , 1 2 , 4
The factors of polynomial are:
( x 4 ) , ( x ( 1 + 2 ) ) , ( x ( 1 2 ) )
The polynomial of degree 3 will be of the form:
f ( x ) = a ( x 4 ) ( x ( 1 + 2 ) ) ( x ( 1 2 ) ) = a ( x 4 ) ( x 1 2 ) ) ( x 1 + 2 ) ) = a ( x 4 ) ( ( x 1 ) 2 ) ) ( ( x 1 ) + 2 ) )
Then
= a ( x 4 ) ( ( x 1 ) 2 ( 2 ) 2 ) ) = a ( x 4 ) ( ( x 1 ) 2 2 ) ) = a ( x 4 ) ( x 2 2 x + 1 2 ) = a ( x 4 ) ( x 2 2 x 1 ) = a ( x ( x 2 2 x 1 ) 4 ( x 2 2 x 1 ) ) = a ( ( x 3 2 x 2 x ) ( 4 x 2 8 x 4 ) ) = a ( x 3 2 x 2 x 4 x 2 + 8 x + 4 ) = a ( x 3 6 x 2 + 7 x + 4 )
Since the given polynomials in options have leading coefficient 1, we shall put a = 1.
f ( x ) = a ( x 3 6 x 2 + 7 x + 4 ) = 1 ( x 3 6 x 2 + 7 x + 4 ) = x 3 6 x 2 + 7 x + 4 f ( x ) = x 3 6 x 2 + 7 x + 4
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