How to write a polynomial function of least degree with integral coefficients that has the given zeros -3, -1/3, 5?

amebulauvbr

amebulauvbr

Answered question

2023-03-16

How to write a polynomial function of least degree with integral coefficients that has the given zeros -3, -1/3, 5?

Answer & Explanation

awesomeamber802x

awesomeamber802x

Beginner2023-03-17Added 8 answers

If the zero is c, the factor is (x-c).
So for zeros of - 3 , - 1 3 , 5 , the factors are
( x + 3 ) ( x + 1 3 ) ( x - 5 )
Let's take a look at the factor ( x + 1 3 ) . Using the factor in this form will not result in integer coefficients because 1 3 is not an integer.
Move the 3 in front of the x and leave the 1 in place: ( 3 x + 1 )
When set equal to zero and solved, both
( x + 1 3 ) = 0 and ( 3 x + 1 ) = 0 result in x = - 1 3
f ( x ) = ( x + 3 ) ( 3 x + 1 ) ( x - 5 )
Multiply the first two factors.
f ( x ) = ( 3 x 2 + 10 x + 3 ) ( x - 5 )
Multiply/distribute again.
f ( x ) = 3 x 3 + 10 x 2 + 3 x - 15 x 2 - 50 x - 15
Combine like terms.
f ( x ) = 3 x 3 - 5 x 2 - 47 x - 15

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