Find the zeros of the polynomial function n(x)=−0.5x3−1.5x2+2x
, given its graph. Enter the zeros in order from least to greatest.
(2 points)
The zeros of n(x)
are x=
, x=
, and x=
.
To find the zeros of the polynomial function, we need to find the values of x for which n(x) equals zero. In other words, we need to find the x-intercepts of the graph of n(x).
We can see from the given function that the graph is a cubic polynomial. Cubic polynomials can have at most three zeros. Therefore, the graph of n(x) will have at most three x-intercepts.
To find the zeros, we can set n(x) equal to zero and solve for x:
-0.5x^3 - 1.5x^2 + 2x = 0
Factoring out an x, we get:
x(-0.5x^2 - 1.5x + 2) = 0
Either x = 0 or -0.5x^2 - 1.5x + 2 = 0.
We can solve the quadratic equation -0.5x^2 - 1.5x + 2 = 0 by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
In this equation, a = -0.5, b = -1.5, and c = 2. Substituting these values into the quadratic formula, we get:
x = (-(-1.5) ± √((-1.5)^2 - 4(-0.5)(2)))/(2(-0.5))
Simplifying,
x = (1.5 ± √(2.25 + 4))/(-1)
x = (1.5 ± √(6.25))/(-1)
x = (1.5 ± 2.5)/(-1)
So, we have two possible solutions:
x = (1.5 + 2.5)/(-1) = 4/(-1) = -4
x = (1.5 - 2.5)/(-1) = -1/(-1) = 1
Therefore, the zeros of n(x) are x = -4 and x = 1.
To find the zeros of the polynomial function, we need to find the values of x for which the function n(x) equals zero.
Given the polynomial function n(x) = -0.5x^3 - 1.5x^2 + 2x, we set n(x) equal to zero and solve for x:
-0.5x^3 - 1.5x^2 + 2x = 0
Factoring out an x, we have:
x(-0.5x^2 - 1.5x + 2) = 0
Now, we have two factors: x = 0 or -0.5x^2 - 1.5x + 2 = 0.
To find the zeros of the quadratic equation -0.5x^2 - 1.5x + 2 = 0, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = -0.5, b = -1.5, and c = 2. Plugging these values into the quadratic formula, we have:
x = (-(-1.5) ± √((-1.5)^2 - 4(-0.5)(2))) / (2(-0.5))
x = (1.5 ± √(2.25 + 4)) / (-1)
x = (1.5 ± √(6.25)) / (-1)
x = (1.5 ± 2.5) / (-1)
This gives us two potential zeros:
x = (1.5 + 2.5) / (-1) = 4 / (-1) = -4
x = (1.5 - 2.5) / (-1) = -1 / (-1) = 1
So, the zeros of the polynomial function n(x) = -0.5x^3 - 1.5x^2 + 2x are x = -4, x = 0, and x = 1.
To find the zeros of the polynomial function n(x) = −0.5x^3 − 1.5x^2 + 2x, we need to set the function equal to zero and solve for x.
Step 1: Set n(x) equal to zero:
−0.5x^3 − 1.5x^2 + 2x = 0
Step 2: Factor out the common factor, if possible:
x(-0.5x^2 - 1.5x + 2) = 0
Step 3: Apply the zero-product property and set each factor equal to zero:
x = 0 (from the first factor)
-0.5x^2 - 1.5x + 2 = 0 (from the second factor)
Step 4: Solve the quadratic equation:
To solve the quadratic equation -0.5x^2 - 1.5x + 2 = 0, we can use factoring, completing the square, or the quadratic formula.
Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = -0.5, b = -1.5, and c = 2. Plugging the values into the quadratic formula, we get:
x = (-(-1.5) ± √((-1.5)^2 - 4(-0.5)(2))) / (2(-0.5))
x = (1.5 ± √(2.25 + 4)) / (-1)
x = (1.5 ± √(6.25)) / (-1)
x = (1.5 ± 2.5) / (-1)
x = (1.5 + 2.5) / (-1) or x = (1.5 - 2.5) / (-1)
x = 4 / (-1) or x = -1 / (-1)
x = -4 or x = 1
So, the zeros of the polynomial function n(x) = −0.5x^3 − 1.5x^2 + 2x are:
x = -4, x = 0, and x = 1.