Question

write the polynomial divisor, dividend, and quotient functions represented by the synthetic division shown below.
-3 | 1 -2 -9 18
| -3 15 -18
----------------
1 -5 6 0

the divisor is | x + 3

the dividend is | x

the quotient is

Explanation:

Step1: Recall Synthetic Division

In synthetic division, the coefficients of the dividend polynomial are used. The top row (1, -2, -9, 18) are coefficients of the dividend. The degree of the dividend is one more than the number of coefficients minus 1? Wait, no: the number of coefficients is 4, so degree is 3. So the dividend is \(x^3 - 2x^2 - 9x + 18\). Let's check: when we do synthetic division with divisor \(x + 3\) (root at -3), the coefficients of dividend are 1 (x³), -2 (x²), -9 (x), 18 (constant).

Step2: Find Dividend

Using the synthetic division setup, the dividend's coefficients are 1 (for \(x^3\)), -2 (for \(x^2\)), -9 (for \(x\)), and 18 (constant term). So the dividend polynomial is \(x^3 - 2x^2 - 9x + 18\).

Step3: Find Quotient

The bottom row (excluding the last 0) are coefficients of the quotient. The coefficients are 1 (x²), -5 (x), -6 (constant). So the quotient is \(x^2 - 5x - 6\). Let's verify: \((x + 3)(x^2 - 5x - 6) = x^3 -5x^2 -6x + 3x^2 -15x -18 = x^3 -2x^2 -21x -18\)? Wait, no, wait my calculation is wrong. Wait the bottom row is 1, -5, -6, 0. Wait the dividend is \(x^3 -2x^2 -9x +18\), divisor \(x +3\). Let's do polynomial division: divide \(x^3 -2x^2 -9x +18\) by \(x +3\).

\(x^3 ÷ x = x^2\). Multiply \(x +3\) by \(x^2\): \(x^3 +3x^2\). Subtract from dividend: \((x^3 -2x^2 -9x +18) - (x^3 +3x^2) = -5x^2 -9x +18\). Next term: \(-5x^2 ÷ x = -5x\). Multiply \(x +3\) by \(-5x\): \(-5x^2 -15x\). Subtract: \((-5x^2 -9x +18) - (-5x^2 -15x) = 6x +18\). Next term: \(6x ÷ x = 6\). Multiply \(x +3\) by 6: \(6x +18\). Subtract: 0. So quotient is \(x^2 -5x +6\)? Wait wait the bottom row in synthetic division is 1, -5, -6, 0. Wait the numbers in synthetic division: the top row is coefficients of dividend: 1 (x³), -2 (x²), -9 (x), 18 (constant). The divisor root is -3. Then we bring down 1, multiply by -3: -3, add to -2: -5. Multiply -5 by -3: 15, add to -9: 6. Multiply 6 by -3: -18, add to 18: 0. Wait wait I misread the numbers! The top row is 1, -2, -9, 18. Then the middle row is -3, 15, -18. Then bottom row is 1, -5, 6, 0. Oh! I misread -9 +15 is 6, not -6. So the bottom row is 1, -5, 6, 0. So the quotient coefficients are 1 (x²), -5 (x), 6 (constant). So quotient is \(x^2 -5x +6\). Let's verify: \((x +3)(x^2 -5x +6) = x^3 -5x^2 +6x +3x^2 -15x +18 = x^3 -2x^2 -9x +18\), which matches the dividend. So my earlier mistake was misreading the middle addition. So the bottom row (excluding last 0) is 1, -5, 6. So quotient is \(x^2 -5x +6\). And dividend is \(x^3 -2x^2 -9x +18\).

Answer:

Dividend: \(x^3 - 2x^2 - 9x + 18\)
Quotient: \(x^2 - 5x + 6\)