Question 935931: Given the function :h(x)=-x^5-2x^4 + x + 2, please answer the following questions:
1- describe the end behavior of this function.
2- Use the rational zero theorem to list the possible rational zeros of h.
3- Use synthetic division with the remainder and factor theorems to find all real roots of h.
4- Graph the function and label any x-intercepts,local minimum(s) and maximum(s).thank you.
Answer by josgarithmetic(39614)   (Show Source):
You can put this solution on YOUR website! Check possible roots, -1, 1, -2, 2, but being degree 5, try to check for one more possible root.
Be sure to use dividend of  when doing the first synthetic division finding the first root.
 leading coefficient for degree 5 means h increases toward the left and decreases toward the right, as x tends unbounded.
(EDIT: narrowing domain and range)
Three rational real roots, and two other complex roots.
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SOME FURTHER HELPFUL DISCUSSION
In case you not yet know HOW to do synthetic division, you can use polynomial division to test the binomial factors of h(x). You still must begin with all terms in the original polynomial, INCLUDING any with coefficients of 0. Aligned with the Rational Roots Theorem, you are testing for x+1, x-1, x+2, x-2, as possible "ROOTS". You WILL find that three of them work as rational real roots.
The question number 3:
When you do the division, if you get remainder of 0, then you found a root. If you get a nonzero remainder, then the root tested is not in fact a root of the function, but represents a point on the function (but just is not on the x-axis); this is the REMAINDER THEOREM. When your tested root or zero gives a remainder of 0, then it is in fact, a root or a zero, and this is what the FACTOR THEOREM tells you.
I did not try to determine the minimum and maximum. You may be able to try derivative (calculus) and equate to zero to find the x-value which correspond. In any case, you should have found using either polynomial or synthetic division, roots are -2, -1, and 1.
(Do you need to see the synthetic division work?)
(Answer = YES)
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THE SYNTHETIC DIVISION
Roots found are -2, -1, 1.
Coefficients of dividend are : -1,-2,0,0,1,2
Polynomial of degree 5.
_________|_______-1____-2___0____0____1_____2_
___-1_____|
_________|____________1_____1____-1____1_____-2
______________________________________________
_________________-1____-1_____1____-1____2____0
Remainder is zero, so -1 is a root.
The last row, bottom, is now the new dividend to look for the next
root.
_________|________-1____-1_____1____-1____2____
____-2___|
_________|_____________2_____-2____2____-2
______________________________________
_________________-1_____1______-1____1_____0_____
remainder zero.
_________|________-1_____1______-1____1_____
____1____|
_________|______________-1_____0_____-1
______________________________________
_________________-1_____0_____-1______0______
remainder zero.
The last quotient means  as a non-factorable quadratic
factor of h(x). The roots, two of them will have imaginary component.
The roots are not Real numbers.
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