A student attempted to divide 4 x^3+2 x^2+6 by x+2 synthetically by setting up the division as

step 1 If you're dividing 8x to the 6 plus 6x cubed by 2x to the 4th, you would get 8x to the 6 over 2x

A student attempted to divide 4 x^3+2 x^2+6 by x+2...

A student attempted to divide
$$
4 x^{3}+2 x^{2}+6 \text { by } x+2
$$
synthetically by setting up the division as follows
$$- 2)\overline{ 4 26}$$
This is incorrect. WHAT WENT WRONG? Give the correct setup and the answer.

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Key Concepts

Synthetic Division

Synthetic division is a simplified method for dividing a polynomial by a linear factor of the form x - c. It uses only the coefficients of the polynomial and the value c (or –c when the divisor is in the form x + a) to efficiently compute the quotient and remainder. This method streamlines the division process compared to long division, but it requires careful attention to the order and completeness of the coefficients.

Including Missing Coefficients

When setting up synthetic division, it is necessary to include coefficients for all descending powers of x, even if some coefficients are zero. For instance, in a polynomial like 4x^3 + 2x^2 + 6, the term for x is missing and should be represented by a 0. Omitting this zero coefficient leads to an incorrect setup and erroneous result, as the synthetic division process depends on the precise order and number of terms.

Using the Correct Root for the Divisor

For synthetic division the divisor must be written as x - c; if the divisor is given in a different form, such as x + 2, it needs to be rewritten correctly by identifying c = -2. Using the correct value ensures that the division process properly accounts for the sign change and computes the accurate quotient and remainder.

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