Use the long division method to find the result when x, cubed, plus, 9, x, squared, plus, 21, x, plus, 9x

Question

3
+9x
2
+21x+9 is divided by x, plus, 3x+3.

Answer 1

To use the long division method to divide the polynomial $x^3+9x^2+21x+9$ by the polynomial $x+3$, we can follow these steps:

Step 1: Set up the long division

We write it in the long division format:

          ____________
x + 3  |  x^3 + 9x^2 + 21x + 9

Step 2: Divide

1. First term: Divide the first term of the dividend $x^3$ by the first term of the divisor $x$: $$\frac{x^3}{x}=x^2$$ Write $x^2$ above the division line.

2. Multiply: Multiply $x^2$ by $x+3$: $$x^2(x+3)=x^3+3x^2$$ Write this product beneath the dividend.

3. Subtract: Subtract this result from the original polynomial: $$(x^3+9x^2)-(x^3+3x^2)=6x^2$$ Bring down the next term $+21x$: $$6x^2+21x$$

Step 3: Repeat the process

1. Next term: Divide the first term $6x^2$ by $x$: $$\frac{6x^2}{x}=6x$$ Write $+6x$ above the division line.

2. Multiply: Multiply $6x$ by $x+3$: $$6x(x+3)=6x^2+18x$$ Write this product beneath $6x^2+21x$ and subtract: $$(6x^2+21x)-(6x^2+18x)=3x$$ Bring down the next term $+9$: $$3x+9$$

Step 4: One more time

1. Next term: Divide $3x$ by $x$: $$\frac{3x}{x}=3$$ Write $+3$ above the division line.

2. Multiply: Multiply $3$ by $x+3$: $$3(x+3)=3x+9$$ Subtract: $$(3x+9)-(3x+9)=0$$

Final Result

Since the remainder is zero, the final result of dividing $x^3+9x^2+21x+9$ by $x+3$ is: $$\boxed{{\displaystyle x^2+6x+3}}$$