Question

100 POINTS PLS HELP
Divide.
4x^3+6x+18/x+2

Answer 1

To divide the polynomial 4x^3 + 6x + 18 by x + 2, use long or synthetic division, dividing each term separately and following the appropriate steps for each method.

Step-by-step explanation:

The division of the polynomial 4x^3 + 6x + 18 by x + 2 can be carried out using long division or synthetic division. In long division, you divide the first term of the numerator by the first term of the denominator (4x^3/x) and multiply the result by the entire divisor, subtract that product from the numerator, and continue the process with the resulting polynomial until you reach a remainder or a term that can no longer be divided by x+2. Synthetic division is a shortcut where you use coefficients and a special set of steps to simplify the process.

The division of exponentials rule is also relevant here; where you divide the coefficients and subtract the exponents when the bases are the same. In this case, there are no like bases, so this rule does not apply directly. Instead, we divide each term separately by the linear divisor.

Answer 2

To perform polynomial long division, we need to divide the highest degree term of the numerator by the highest degree term of the denominator. In this case, that means dividing 4x^3 by x, which gives us:

4x^3 / x = 4x^2

So our first term in the quotient is 4x^2. We then multiply the divisor (x+2) by 4x^2 to get:

4x^2(x+2) = 4x^3 + 8x^2

We then subtract this from the numerator to get the remainder:

(4x^3 + 6x + 18) - (4x^3 + 8x^2) = -8x^2 + 6x + 18

Since the degree of the remainder (-8x^2 + 6x + 18) is less than the degree of the divisor (x+2), we cannot divide any further. Therefore, our final answer is:

4x^2 - 8x + 6 + (18 / (x+2))

So we have:

4x^3 + 6x + 18 / x + 2 = (4x^2 - 8x + 6) + (18 / (x+2))