Thread: Please help me with this sample test

Hello everybody, could you please give me a hand with this sample test

1-form a polynomial whose zeros and degree are given

zeros : -1, 1, 3; degree 3

2-for the polynomial function, a- list each zero and its multiplicity, b= determine whether the graph croses or touches the x axis at each x intercept. C find the power function that the graph of f resembles for large values of |x|

f(x)= (x-5)^3 (x+4)^2

3- for the given polynomial function f:

a-find the x and y intercepts of f
b-determine whether the graph of f crosses or touches the x axis at each x intercept
c-end behavior. Find the poer funcion that the graph of f resembles for large values of |x|
d- determine the maximum number of turning points on the graph of f
e- Use the x intercepts to find the intervals on which the graph of f is above and below the x axis
f- plot the points obtained in parts a- and e- and use the remaining information to conect them with a smooth, continuos curve

F(x)= (x-1)(x-2)(x-4)


4- find the vertical, horizontal and oblique asymptopes, if any, in the given rational function

3x+5

5-find the vertical, horizontal and oblique asymptotes, of the given rational function

f(x)-=3x+5/x-6

6-Use the factor theorem to determine whether x-c is a factor of f(x)

f(x) = -4x^7+x^3-x^2+2


7-List the potential zeros of each polynomial function
f(x)=6x^4-x^2+9

8-Use Descartes' rule of signs and the rational zeros theorem to find all real zeros in the given polynomial functions, use the zeros to factor f over the real numbers

f(x)= X^3+2x^2-5x-6
f(x)=2x^3+x^2+2x+1

9-Solve the given equation in the real number system

2x^3-3x^2-3x-5=0

10-
Use the intermediate value theorem to show that given polynomial function has a zero in the given interval

f(x) = 8x^4-2x^2+5x-1 {0,1}

11-In the following problem information is given about the polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f

Degree 4; zeros: i,1+i


12-Form a polynomial f(x) with real coefficients having the given degree and zeros

Degree 5; zeros: 2, -i;1+i

13-Use the given zero to find the remaining zeros of the given function

F(x) = 2x^4+5x^3+5x^2+20x-12 ; Zero: -2i
14-
Find the complex zeros of the given polynomial function. Write f in factored form

F(x) = x^4+5x^2+4


Thank you so much.

Originally Posted by jhonwashington

1-form a polynomial whose zeros and degree are given

zeros : -1, 1, 3; degree 3

This will be a cubic, so (x-(-1))(x-1)(x-3) is one such cubic, just multiply
out the brackets.

RonL

Originally Posted by jhonwashington

13-Use the given zero to find the remaining zeros of the given function

F(x) = 2x^4+5x^3+5x^2+20x-12 ; Zero: -2i

As the polynomial has real coefficients the complex roots exist as conjugate
pairs, so if x=-2i is a root so is x=2i, which means that (x-2i)(x+2i)=x^2+4 is
a factor of F(x).

So dividing F(x) by x^2+4, we find:

2x^4+5x^3+5x^2+20x-12 = (x^2+4)(2x^2+5x -3)

Now use the quadratic formula to find the roots of 2x^2+5x -3, which are
x=1/2 and x=-3.

Thus the roots of F(x) are -2i, 2i, 1/2 and -3 (which we know is a complete list
of the roots of F(x), as every polynomial of degree n has at most n complex roots).

RonL

Hello, jhonwashington!

7) List the potential zeros of this polynomial function: .f(x) .= .6x^4 - x^2 + 9

A potential zero has the form: n/d
. . where n is a factor of the constant term
. . and d is a factor of the leading coefficient.

The factors of 9 are: ±1, ±3, ±9
The factors of 6 are: ±1, ±2, ±3, ±6

The potential zeros are: ±1, ±3, ±9, ±1/2, ±3/2, ±9/2, ±1/3, ±1/6

9) Solve the given equation in the real number system: .2x³ - 3x² - 3x - 5 .= .0

Using the Remainder Theorem, we find that x = 5/2 is a zero.
. . Hence, (2x - 5) is a factor.

Then we have: .(2x - 5)(x² + x + 1) .= .0

The quadratic factor has no real roots.

Therefore, the only real root is: .x = 5/2

12) Form a polynomial f(x) with real coefficients having the given degree and zeros:
. . . Degree 5; zeros: 2, -i; 1 + i

Complex roots always appear in conjugate pairs.
If i is a zero, then -i is a zero.
If 1 + i is a zero, then 1 - i is a zero.

The polynomial is: .(x - 2) (x - i) (x + i) (x - [1 + i]) (x - [1 - i])

I'll let you multiply it out . . .

14) Find the complex zeros of the given polynomial function. Write F in factored form.
. . F(x) .= .x^4 + 5x^2 + 4

We have: .x^4 + 5x^2 + 4 .= .0

Factor: .(x² + 1)(x² + 4) .= .0

Then: .x² + 1 .= .0 . → . x² = -1 . → . x = ±i
. . . . . x² + 4 .= .0 . → . x² = -4 . → . x = ±2i

. . F(x) .= .(x - i)(x + i)(x - 2i)(x + 2i)