What is the quotient of the polynomial 2x^3 + 2x + 12 when using synthetic division with x^2 as the divisor?

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Synthetic division is a simplified method of dividing polynomials, particularly useful when dividing by linear factors. However, since we are dividing by a polynomial of higher degree (specifically, x² in this case), we can still use a process similar to synthetic division, though it requires more attention to coefficients.

First, let’s express the polynomial properly:

Our polynomial is:

2x^3 + 0x^2 + 2x + 12

Now, we can set up the division:

   2x^3 + 0x^2 + 2x + 12 
/  (x^2 + 0x + 0)

We start by taking the leading term of the dividend (2x³) and divide it by the leading term of the divisor (x²):


    2x^3 ÷ x^2 = 2x

We then multiply the entire divisor by this result and subtract it from the original polynomial:

   2x^3 + 0x^2 + 2x + 12 
- (2x * (x^2 + 0x + 0))   
---------------------------  
   0 + 2x + 12

Next, we proceed with the remainder, which is now:

2x + 12

Next, we divide 2x by x² (noting that we cannot actually divide a linear term by a quadratic polynomial, which indicates that the division is complete). Therefore, the quotient we have is:

2x

As for the remainder, we denote it as:

(2x + 12)/(x^2)

In summary, the result of the synthetic division gives us a quotient of:

Quotient: 2x

Remainder: (2x + 12)/(x^2)

This shows that the polynomial is not divisible evenly by x², leaving us with this remainder.

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