fill in the missing values below one at a time to find the quotient when $-x^{3}-3x^{2}+4$ is divided by $x…

fill in the missing values below one at a time to find the quotient when $-x^{3}-3x^{2}+4$ is divided by $x + 2$.

fill in the missing values below one at a time to find the quotient when $-x^{3}-3x^{2}+4$ is divided by $x + 2$.

Answer

Explanation:

Step1: Analyze the coefficient relationship

We know that when we multiply (x) by the term in the top - middle cell (let's call it (ax)) to get (-x^{2}). Using the rule of polynomial multiplication ((x)\times(ax)=ax^{2}). Since (ax^{2}=-x^{2}), then (a = - 1).

Step2: Fill in the other cells

For the cell below (-x^{2}) (in the (+2) row and (-x) column): ((+2)\times(-x)=-2x) For the last cell (constant term): The original polynomial is (-x^{3}-3x^{2}+4). The sum of the products from the table: ((-x^{3}-x^{2}-2x^{2}-2x)+4). Combining like terms (-x^{3}-3x^{2}-2x + 4). The last cell (constant term) is (4) (because when we divide (-x^{3}-3x^{2}+4) by (x + 2), we can also think of it as ((x + 2)(-x^{2}-x)+4))

Answer:

The missing value in the top - middle cell is (-x), the value in the (+2) row and (-x) column is (-2x), and the value in the last cell (constant term) is (4)