Exam 13: Polynomial and Synthetic Division

If x = $\frac { 2 } { 7 }$ is a root of $49 x ^ { 3 } - 126 x ^ { 2 } + 60 x - 8 = 0$ ,use synthetic division to factor the polynomial completely and list all real solutions of the equation.

Find all the real zeros of $f ( x ) = 2 x ^ { 3 } - 6 x ^ { 2 } + 2 x - 6$ .

If x = -2 is a root of $x ^ { 3 } + 5 x ^ { 2 } - 4 x - 20 = 0$ ,use synthetic division to factor the polynomial completely and list all real solutions of the equation.

Select the correct graph of the following function.​ $f ( x ) = 5 x ^ { 3 } - x ^ { 2 } - 12 x + 7$

Use synthetic division to perform the division.​ $\frac { 3 x ^ { 3 } - 6 x ^ { 2 } - 23 x + 8 } { x + 2 }$ ​

Let $P ( x ) = 2 x ^ { 3 } + 9 x ^ { 2 } - 7 x + 3$ . ​ Use synthetic division to find the value P (-7).

Use synthetic division to divide. $\left( 4 x ^ { 3 } + 11 x ^ { 2 } - 43 x + 10 \right) \div ( x + 5 )$

The amounts A (in billions of dollars)donated to support higher education in the United States from 2000 through 2007 are shown in the table,where t represents the year,with t = 0 corresponding to 2000. Year, t Amount, A 0 23.3 1 24.3 2 24 3 24 4 24.5 5 25.7 6 28.1 7 29.9 Use a graphing utility to select a correct a scatter plot of the above data.

Use synthetic division to divide.​ $\frac { 4 x ^ { 3 } + 20 x ^ { 2 } - 27 x - 18 } { x + \frac { 1 } { 2 } }$ ​

Use the Remainder Theorem and synthetic division to find the function value.Verify your answers using another method.​ $h ( x ) = x ^ { 3 } - 2 x ^ { 2 } - 5 x + 6 , \quad h ( - 8 )$ ​

Find all the real zeros of $f ( x ) = 4 x ^ { 3 } - 20 x ^ { 2 } + 17 x - 4$ .

Use Descartes' Rule of Signs to determine the possible number of positive and negative zeros of $f ( x ) = 4 x ^ { 3 } + 3 x ^ { 2 } + 5 x + 1$ .

Use synthetic division to divide.​ $\frac { x ^ { 3 } - 125 } { x - 5 }$ ​

Write $f ( x ) = x ^ { 3 } + 7 x ^ { 2 } + 13 x + 22$ in the form $f ( x ) = ( x - k ) q ( x ) + r$ when k = -5.

Use long division to divide.​ $\left( x ^ { 4 } + 4 x ^ { 2 } + 6 \right) \div \left( x ^ { 2 } + x + 4 \right)$

Use long division to divide.​ $\left( x ^ { 3 } - 343 \right) \div ( x - 7 )$ ​

Use Descartes' Rule of Signs to determine the possible number of positive and negative zeros of $f ( x ) = 2 x ^ { 3 } - 2 x ^ { 2 } + 3 x - 4$ .

Use the Remainder Theorem and synthetic division to find the function value.​ $g ( x ) = 3 x ^ { 6 } + 4 x ^ { 4 } - 6 x ^ { 2 } + 4 , \quad g ( 0 )$ ​

Using the factors (x - 5)and (x + 2),find the remaining factor(s)of $f ( x ) = x ^ { 3 } - 6 x ^ { 2 } - x + 30$ and write the polynomial in fully factored form.

Use the zero or root feature of a graphing utility to approximate the zeros of $f ( x ) = 2 x ^ { 3 } + 8 x ^ { 2 } - 5 x - 7$ accurate to the nearest thousandth.