  ### Home > PC > Chapter 9 > Lesson 9.3.2 > Problem9-116

9-116.

Show whether or not $x − 5$ is a factor of $x^{3} + 4x^{2} − 11x − 30$.

Set up the generic rectangle with the divisor on the left of the box. Input only the first terms of the dividend. Generic rectangle, 2 rows, 4 columns, left edge labeled, x minus 5, interior top left, labeled x cubed.

Since $x$ divides into $x^3$, $x^2$ times, write the $x^2$ on top and multiply $x^2$ with the $−5$ and fill in the grid with the product. Labels added, top edge left, x squared, interior bottom left, negative 5, x squared.

Add an expression to $−5x^2$ in order to get the needed $4x^2$. Add it to the right of $x^3$Label added, interior top, second from left, 9 x squared.

Dividing $x$ into $9x^2$ gives $9x$. Add the $9x$ at the top. Then multiply $9x$ and $−5$ and fill the answer in the grid. Labels added: top edge, second from left, + 9 x, interior bottom, second from left, negative 45 x.

Add an expression to $−45x$ in order to get the needed $−11x$. Add it to the right of $9x^2$Label added: interior top, second from right, 34 x.

Dividing $x$ into $34x$ gives $34$. Add the $34$ at the top. Then multiply $34$ and $−5$ and fill the answer in the grid. Labels added: top edge, second from right, + 34, interior bottom, second from right, negative 170.

Add an expression to $−170$ in order to get the needed $−30$. Add it to the right of $34x$. This is the remainder. Add it to the top as a ratio. Labels added: top edge, right, 140 divided by quantity x minus 5. Interior top right, 140.       Since there is a remainder, $x − 5$ is not a factor!