Question

divide using synthetic division.

\\((6x^2 + x - 7) \div (x - 1)\\)

\\((6x^2 + x - 7) \div (x - 1) = \\)
(simplify your answer. use integers or fractions for any numbers in the expression. do not factor.)

Explanation:

Identify the division setup

We need to divide the polynomial \(6x^2 + x - 7\) by the binomial \(x - 1\) using synthetic division.

• The dividend is \(6x^2 + x - 7\), with coefficients: \(6\), \(1\), and \(-7\).
• The divisor is \(x - 1\). For synthetic division, we use the zero of the divisor, which is \(c = 1\).

Perform synthetic division

We set up the synthetic division table with \(1\) on the outside and the coefficients \(6\), \(1\), \(-7\) on the inside:

  1 |   6    1   -7
    |        6    7
    ----------------
        6    7    0

• Bring down the first coefficient: \(6\).
• Multiply \(1 \times 6 = 6\), and write it under the next coefficient \(1\).
• Add: \(1 + 6 = 7\).
• Multiply \(1 \times 7 = 7\), and write it under the constant term \(-7\).
• Add: \(-7 + 7 = 0\).

Interpret the quotient and remainder

The bottom row of our synthetic division table gives the coefficients of the quotient and the remainder:

• The coefficients of the quotient are \(6\) and \(7\). Since we divided a degree 2 polynomial by a degree 1 polynomial, the quotient is of degree 1: \(6x + 7\).
• The remainder is \(0\).

Therefore, the result of the division is:

$$ (6x^2 + x - 7) \div (x - 1) = 6x + 7 $$

Answer:

\((6x^2 + x - 7) \div (x - 1) =\) <blank>\(6x + 7\)</blank>