1. Which binomial is a factor of x^3-3x^2-28x?
A. x-4
B.x^2-4
C.x-7***
x^2-7
2. A company's profit is described by the equation. P(x)=-5^2+300x+15,000. Where x is the price in dollars that the company charges for its product. What should the company charge for the product to generate maximum product?
A. 20
B. 30
C. 40
D. 50
The last one I do not get. If I could get help that would be great. Thank you.
The last one I got is B. 30 dollars
x = -b/2a
x = -300/2(-5)
x = -300/-10
x = 30
both are correct
To determine which binomial is a factor of the polynomial x^3 - 3x^2 - 28x, we can use polynomial factorization or synthetic division.
Let's test each binomial option by using synthetic division:
A. x - 4:
We write the coefficients of the polynomial: 1, -3, -28, 0.
Then we divide the polynomial by x - 4:
4 | 1 -3 -28
| 4 4
|________________
1 1 -24
Since the remainder is not zero, x - 4 is not a factor of x^3 - 3x^2 - 28x.
B. x^2 - 4:
We write the coefficients of the polynomial: 1, -3, -28, 0.
Then we divide the polynomial by x^2 - 4:
4 | 1 -3 -28 0
| 4 4 16
|_________________
1 1 -24 -12
Again, the remainder is not zero, so x^2 - 4 is not a factor of x^3 - 3x^2 - 28x.
C. x - 7:
We write the coefficients of the polynomial: 1, -3, -28, 0.
Then we divide the polynomial by x - 7:
7 | 1 -3 -28
| 7 28
| _______________
1 4 0
In this case, the remainder is zero. Therefore, x - 7 is a factor of x^3 - 3x^2 - 28x.
Hence, the correct answer is C. x - 7.
Now, for the second question, to find the price that generates the maximum profit, we need to determine the x-value at the vertex of the quadratic function P(x) = -5x^2 + 300x + 15000.
We know that the x-value of the vertex of a quadratic function in the form of ax^2 + bx + c is given by the formula x = -b/2a.
In this case, a = -5 and b = 300, so we substitute these values into the formula:
x = -(300) / (2 * (-5))
x = -300 / -10
x = 30
Therefore, the company should charge $30 for its product to generate the maximum profit.
The correct answer is B. 30.