Question

use the long division method to find the result when $2x^3 + 21x^2 + 28x + 9$ is divided by $2x + 1$.

Explanation:

Step1: Divide leading terms

$\frac{2x^3}{2x} = x^2$
Multiply divisor by $x^2$: $(2x+1)x^2 = 2x^3 + x^2$
Subtract from dividend:
$(2x^3 + 21x^2 + 28x + 9) - (2x^3 + x^2) = 20x^2 + 28x + 9$

Step2: Divide new leading terms

$\frac{20x^2}{2x} = 10x$
Multiply divisor by $10x$: $(2x+1)10x = 20x^2 + 10x$
Subtract from current polynomial:
$(20x^2 + 28x + 9) - (20x^2 + 10x) = 18x + 9$

Step3: Divide new leading terms

$\frac{18x}{2x} = 9$
Multiply divisor by $9$: $(2x+1)9 = 18x + 9$
Subtract from current polynomial:
$(18x + 9) - (18x + 9) = 0$

Answer:

$x^2 + 10x + 9$