What is thesum of the two polynomials ? (-x^2+9)+(-3x^2-11x+4)

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Firlakuza [10]

Answer:

Domain: All the real values except x = 2 and x = 4: R - {2, 4}
Holes: x = 2
VA, vertical asymptores: x = 4
HA: horizontal asymptotes: there are not horizontal asymptotes
OA: oblique asymptotes: x + 6 [note that OH does not stand for any known feature, and so it is understood that it was intended to write OA]
Roots: x = 2
Y-intercept: -1

Step-by-step explanation:

1. Given:

$f(x)=\frac{x^3-8}{x^2-6x+8}$

Note that the number 8 in the numerator is not part of the power.

Type of function: rational function

2. Domain: is the set of x-values for which the function is defined.

The given function is defined for all x except those for which the denominator equals 0.

Denominator: x² -6x + 8 = 0

Solve for x:

Factor. (x - 4 )(x - 2) = 0

Zero product property: (x - 4) = 0 or (x - 2) = 0

x - 4 = 0 ⇒ x = 4

x - 2 = 0 ⇒ x = 2

Domain:

All the real values except x = 2 and x = 4: x ∈ R / x ≠ 2 and x ≠ 4.

3. Holes:

The holes on the graph of a rational function are at those x-values for which both the numerator and denominator are zero.

Find the values for which the numerator is zero:

Numerator: x³ - 8 = 0

Factor using difference of cubes property:

a³ - b³ = (a - b)(a² + ab + b²)

x³ - 8 = (x - 2)(x² + 2x + 4) = 0

Zero product property: (x - 2)(x² + 2x + 4) = 0

x - 2 = 0 ⇒ x = 2

x² + 2x + 4 = 0 (this has not real solution)

The values for which the denominator is zero were determined above: x = 2 and x = 4.

Conclusion: for x = 2 both numerator and denominator equal 0, so this is a hole.

4. VA: Vertical asymptotes.

The vertical asymptotes on the graph of a rational function are the vertical lines for which only the denominator (and not the numerator) equals zero.

In the previous part it was determined that happens when x = 4.

5. HA: Horizontal asymptotes.

In rational functions, if the numerator is a higher degree polynomial than the denominator, there is no horizontal asymptote.

6. OA: oblique asymptotes

Find the quotient and the remainder.

x + 6

_______________

x² - 6x + 8 ) x³ + 0x² + 0x - 8

- x³ + 6x² - 8x

___________

6 x² - 8x - 8

- 6x² + 36x - 48

_____________

28x - 56

Result: (x + 6) + (28x - 56) / (x² - 6x + 8)

Find limit x → ∞

$\lim_{x \to \infty}(x + 6) + \frac{28x-56}{x^2-6x+8}=x+6$

7. Roots:

Roots are the values for which f(x) = 0.

That happens when the numerator equals 0, and the denominator is not 0.

As determined earlier: x³ - 8 = 0 ⇒ x = 2.

8. Y-Intercept

The y-intercepts of any function are the y-values when x = 0

Substitute x = 0 into the function:

$f(x)=\frac{x^3-8}{x^2-6x+8}=\frac{0^3-8}{0^2-6(0)+8}}=\frac{-8}{8} =-1$

3 0

4 years ago

BRAINLIEST IF CORRECT 40 POINTS

kati45 [8]

Answer:

D. The checking account is overdrawn by $15.

Step-by-step explanation:

Since the balance on a checking account represents the remaining money left in that checking account, having a negative value for that would mean an overdrawing of money, or drawing more money than is in that account in the first place.

Therefore, the answer would be that the checking account is overdrawn by $15.

Hope this helped!

4 0

3 years ago

Find the slope of the line.

andrey2020 [161]

Answer:

-1

Step-by-step explanation:

To find the slope of a line, we can do rise over run, or y/x.

r/r = 1/1 = 1

But as you can see, the line is going down which means it's decreasing, which means that it has a negative slope

slope = -1

Hope this helped a little!

Have a supercalifragilisticexpialidocious day!

5 0

3 years ago

Simplify (5^3)^6 leaving your answer in index form

Verizon [17]

$( {5}^{3} ) ^{6} = {5}^{3 \times 6} = {5}^{18}$

5 0

3 years ago

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there are 6 pure spectral color : red,orange,yellow, green,blue, and violet. some animals cannot see orange or red. what fractio

Gnoma [55]

Bees see “primary colors” as blue, green and ultra violet. They can distinguish yellow, orange, blue-green, violet, purple, as combinations of their three primary colors.

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3 years ago