Question

divide using synthetic division (2x³ + 4x² - 5) by (x + 3). options: 2x² + 2x - 4 r(-12); 2x² - 2x + 6 r(-23); 2x² - 2x + 8 r(-20); 2x² - 4x + 1 r0

Explanation:

Step1: Identify root & coefficients

Root of divisor $x+3$ is $-3$. Coefficients of dividend $2x^3+4x^2+0x-5$ are $2, 4, 0, -5$.

Step2: Bring down leading coefficient

$\downarrow \quad 2$

Step3: Multiply & add to next coeff

Multiply $2 \times (-3) = -6$. Add to $4$: $4 + (-6) = -2$

Step4: Repeat multiply & add

Multiply $-2 \times (-3) = 6$. Add to $0$: $0 + 6 = 6$

Step5: Final multiply & add

Multiply $6 \times (-3) = -18$. Add to $-5$: $-5 + (-18) = -23$

Step6: Form quotient & remainder

Quotient coefficients: $2, -2, 6$ → $2x^2-2x+6$. Remainder: $-23$

Answer:

$2x^2 - 2x + 6$ R(-23)