Find the rational zeros and then other zeros of the polynomial function f(x) = x^{3} - 17x^{2} + 55x + 25, that is, solve f(x) = 0 Factor f(x) into linear factors

Dolly Robinson 2020-10-27 Answered
Find the rational zeros and then other zeros of the polynomial function f(x)=x3  17x2 + 55x + 25,
that is, solve f(x)=0
Factor f(x) into linear factors
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Expert Answer

cyhuddwyr9
Answered 2020-10-28 Author has 90 answers

To find the rational zero and other zeros of the polynomial
f(x)=x3  17x2 + 55x + 25
Factor f(x) into linear factors
Let us solve the equation.
f(x)=0
x3  17x2 + 55x + 25=0
the constant term has factors 1 and 5.
Let us whether these can be zeros.
f(1)=1  17 + 55 + 25=64  0
 x=1 can not be zero of f(x).
f(5)=53  17(5)2 + 55(5) + 25=125  425 + 275 + 25= 300 + 300=0
 x=5 is a rational zero of f(x).
We can write,
f(x)=x2(x  5) 12x(x  5)  5(x  5)
=(x  5)(x2  12x  5)
Now we only need to solve the quadratic equation.
x2  12x  5=0
Then x=(12) ± (12)2 4(1)(5)2(1)
=12 ± 144 + 202
=12 ± 1642
=12 ± 2412
=6 ± 41
Therefore, there is only one rational zero, 5 and other zeros are
6 + 41, 6  41
The factorization of f(x),
f(x)=(x  5)(x  (6  41))(x  (6 + 41))
=(x  5)(x  6 + 41)(x  6  41)

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Jeffrey Jordon
Answered 2021-11-03 Author has 2047 answers

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