Determine the x-intercepts of the polynomial function y=-6(x+7)^4 + 5. Round to two decimal places, if necessary.
The x axis intercepts are where y = 0
0 = -6(x+7)^4 + 5
(x+7)^4 = 5/6 = 0.8333 ....
(x+7)^2 = +/- 0.91287
if (x+7)^2 = + 0.91287
then
x+7 = +/- 0.977468
or
x = -7.977 or -6.0225
That is two of the four
now if
(x+7)^2 = -0.91287
then
(x+7) = +i sqrt 0.91287
and
(x+7) = -i sqrt 0.91287
so
x = -7 +/- 0.955 i
To determine the x-intercepts of the polynomial function y = -6(x + 7)^4 + 5, we need to set y equal to zero and solve for x.
So, we have 0 = -6(x + 7)^4 + 5.
First, let's isolate the term with the polynomial.
6(x + 7)^4 = 5.
Then, divide both sides by 6.
(x + 7)^4 = 5/6.
Next, take the fourth root of both sides.
x + 7 = ±(5/6)^(1/4).
Now, subtract 7 from both sides.
x = -7 ± (5/6)^(1/4).
This gives us the two x-intercepts of the polynomial function.
Rounding to two decimal places, the x-intercepts are approximately:
x ≈ -7 + 0.78 ≈ -6.22.
x ≈ -7 - 0.78 ≈ -7.78.
Therefore, the x-intercepts of the polynomial function y = -6(x + 7)^4 + 5, rounded to two decimal places, are approximately -6.22 and -7.78.
To determine the x-intercepts of a polynomial function, we need to find the values of x where the y-value is equal to zero. In other words, we need to solve the equation:
-6(x+7)^4 + 5 = 0
First, let's isolate the variable term:
-6(x+7)^4 = -5
Now, divide both sides of the equation by -6:
(x+7)^4 = 5/6
To eliminate the fourth power, we can take the fourth root of both sides:
∛((x+7)^4) = ±√(5/6)
Since we are looking for the x-intercepts, we are only interested in the real solutions. Now, we can simplify the equation:
x + 7 = ±∛(5/6)
Next, subtract 7 from both sides:
x = -7 ±∛(5/6)
Now, we can evaluate the expression -7 ±∛(5/6) to find the two x-intercepts. Rounding to two decimal places, we have:
x ≈ -7 + ∛(5/6) and x ≈ -7 - ∛(5/6)
These values represent the x-coordinates of the points where the graph of the function intersects the x-axis.