First, determine your divisor, which will be the zero of the factor you're given. (It may or may not be a zero, but I'll discuss that later.) In this case, you are dividing by x+1/2, so the divisor is -1/2. I put this in the box on the left.
![fleur]()
Next, bring down the coefficients of the dividend. Since the dividend is 4x^3 + 16x^2 - 23x - 15, the coefficients are 4, 16, -23, and -15.
Now, bring down the first coefficient, 4, and place it below the line. Multiply 4 by the divisor, -1/2, to get -2 and put it below the second coefficient, 16. Now add 16 to -2 to get 14. Repeat these steps until you reach the last coefficient.
4 * -1/2 = -2 -> 16 + -2 = 14
14 * -1/2 = -7 -> -23 + -7 = -30
-30 * -1/2 = 15 -> -15 + 15 = 0
In the end, we have four numbers: 4, 14, -30, 0. These are the coefficients for the quotient. Since the problem asked us to divide 4x^3 + 16x^2 - 23x - 15 (cubic) by x+1/2 (linear), the quotient will be quadratic. Thus, the degree of the first term will be 2, the degree of the second term will be 1, the degree of the third term will be 0 (the third term is a constant), and the last term will be the remainder.
color(white)I
Since the last term in this case is 0, there is no remainder. This tells us that x+1/2 is a factor of 4x^3 + 16x^2 - 23x - 15 because it divides evenly. If there was a remainder, the x=-1/2 would not be a zero, and x+1/2 would not be a factor.
In conclusion, our quotient is color(blue)(4x^2 + 14x -30).
