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How to use synthetic division with x minus c

Use c, not the constant term of the divisor

Synthetic division applies directly to a monic linear divisor $x - c$ . List every coefficient of the dividend, including zeros. Bring down the first coefficient, then repeatedly multiply by $c$ and add in the next column. The final entry is the remainder; the preceding entries are the quotient coefficients.

Problem

Use synthetic division to simplify:

1. $(2 x^{2} + 30 x + 5) \div (x + 7)$ . 2. $(2 x^{2} - 20 x + 50) \div (x - 5)$ . 3. $(- 5 + 7 x + 5 x^{3} - 7 x^{4}) \div (x + 1)$ .

Answer

2 x + 16 - \frac{107}{x + 7}

. 2.

2 x - 10

. 3.

- 7 x^{3} + 12 x^{2} - 12 x + 19 - \frac{24}{x + 1}

The original expressions require x≠-7, x≠5, and x≠-1, respectively.

Step-by-step solution

1. Divide by x plus seven

Since $x + 7 = x - (- 7)$ , use $c = - 7$ , not $7$ . The coefficients are $2, 30, 5$ .

- Bring down $2$ . - Multiply: $- 7 \cdot 2 = - 14$ . Add: $30 + (- 14) = 16$ . - Multiply: $- 7 \cdot 16 = - 112$ . Add: $5 + (- 112) = - 107$ .

The completed bottom row is $2, 16, - 107$ . The quotient is $2 x + 16$ and the remainder is $- 107$ :

2 x^{2} + 30 x + 5 = (x + 7) (2 x + 16) - 107.

If a worksheet asks for the two missing entries in the last column, they are $- 112$ above the line and $- 107$ below it.

2. Divide by x minus five

Use $c = 5$ and coefficients $2, - 20, 50$ . Bring down $2$ , multiply by 5 to get $10$ , and add to $- 20$ to get $- 10$ . Multiply again to get $- 50$ , then add to $50$ to get zero. The bottom row is $2, - 10, 0$ :

2 x^{2} - 20 x + 50 = (x - 5) (2 x - 10) .

The remainder is zero, so the quotient is exactly $2 x - 10$ wherever the original fraction is defined.

3. Reorder a quartic before dividing

Write the third dividend as $- 7 x^{4} + 5 x^{3} + 0 x^{2} + 7 x - 5$ . For $x + 1$ , use $c = - 1$ and coefficients $- 7, 5, 0, 7, - 5$ .

Bring down $- 7$ . The next additions are

5 + 7 = 12, 0 - 12 = - 12, 7 + 12 = 19, - 5 - 19 = - 24.

The quotient is $- 7 x^{3} + 12 x^{2} - 12 x + 19$ and the remainder is $- 24$ .

4. Check the remainder

The remainder theorem gives $R = P (c)$ . Substitution gives $P (- 7) = - 107$ , $P (5) = 0$ , and $P (- 1) = - 24$ for the three respective dividends. These agree with the last entries in the synthetic divisions.

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