Question

Use the long division method to find the result when x^3+7x^2+12x+6x 3 +7x 2 +12x+6 is divided by x+1x+1. If there is a remainder, express the result in the form q(x)+\frac{r(x)}{b(x)}q(x)+ b(x) r(x) ​ .

Answer 1

By long division (x³ + 7·x² + 12·x + 6) ÷ (x + 1) gives the expression;

`x^2 + 5 \cdot x + 7 - (1)/((x + 1))`

Explanation:

The polynomial that is to be divided by long division is x³ + 7·x² + 12·x + 6

The polynomial that divides the given polynomial is x + 1

Therefore, we have;

`\ \ \ \ \ \ \ \ \ \ \ \ x^2 + 5\cdot x + 7\\ (x + 1) √(x^3 + 7\cdot x^2 +12\cdot x + 6) \\\ {} \ {} \ {} \ \ {} \ {} \ {} \ {} \ {} \ {} \ {} \ \ x^3 + 2 \cdot x^2 \\\ \ \ \ {} \ \ {} \ {} \ {} \ \ {} \ {} \ \ \ {} \ {} \ {} \ \ {} \ {} \ {} \ \ 5 \cdot x^2 + 12\cdot x + 6\\ \ {} \ {} \ {} \ {} \ {} \ {} \ \ {} \ {} 5 \cdot x^2 + 5\cdot x\\\ 7\cdot x+6\\7\cdot x+7\\-1`

(x³ + 7·x² + 12·x + 6) ÷ (x + 1) = x² + 5·x + 7 Remainder -1

Expressing the result in the form
`q(x) + (r(x))/(b(x))`, we have;

`(x^3 + 7\cdot x^2 + 12 \cdot x + 6)/ (x + 1) = x^2 + 5 \cdot x + 7 - (1)/((x + 1))`