Answer:
x=-1.334
Step-by-step explanation:
Let first use descrates rule of signs to find possible zeroes.
First, find the number of signs changes to find possible positive zeroes.
Here,
we have
[tex]x {}^{5} - {x + 6}[/tex]
We have 2 signs changes so we have either
2 positive zeroes, or none.
To find. negative subsitue -x for x.
[tex] ( - x) {}^{5} - ( - x) + 6[/tex]
[tex] - x {}^{5} + x + 6[/tex]
There is one sign change so we have 1 negative zzeroes.
The fundamental theorem algebra tells us that for a function which a leading degree n has exactly n roots at most.
The leading degree has 5 roots so we have either
2 positive zeroes 1 positive zero, or 2 complex zeroes.
or
4 complex zeroes, 1 positive zero.
So let see first.
[tex]x {}^{5} - {x}^{2} + 6[/tex]
Rational Roots Theroem isn't applicable, because we have no zeroes. that are rational.
We must use a table of values to find a zero.
If you plug in -2 and -1, the sings oscillate so we have a zero between
-2 and-1.
[tex]f( - 2) = { - 2}^{5} - ( - 2) {}^{2} + 6 = - 30[/tex]
[tex]f( - 1) = - 1 {}^{5} - ( - 1) {}^{2} + 6 = 4[/tex]
So using the Theorem, the zero has to between -30 and 4.
In fact, if we plug the function in a graphing calculator, we get this.
So we have one real zero x=-1.334
