Question 1

Find the quotient and remainder of $\frac { x ^ { 5 } - 3 x ^ { 2 } + 2 x - 2 } { x + 1 }$ .

Accepted Answer

Quotient $$x ^ { 4 } - x ^ { 3 } + x ^ { 2 } - 4 x + 6$$ and remainder $$- 8$$

Question 2

Use synthetic division and the remainder theorem to evaluate $p ( 3 )$ for $p ( x ) = 2 x ^ { 3 } - 2 x ^ { 2 } + 11 x - 100$ .

Accepted Answer

$p ( 3 ) = - 31$ $\begin{array} { r r r r r } 
3 & 2 & - 2 & 11 & - 100 \
\hline && 6 & 12 & 69 \
\hline &2 & 4 & 23 & - 31
\end{array}$

Question 3

Find the maximum or minimum value of the function $f ( x ) = x ^ { 2 } - 4 x + 2$ .

Accepted Answer

$f ( x ) = x ^ { 2 } - 4 x + 2$ $\Rightarrow$ $$a = 1$$ and $b = - 4$ , so the minimum value is $f ( 2 ) = - 2$ .

Question 4

Find all rational zeros of the polynomial $p ( x ) = 15 x ^ { 3 } - 29 x ^ { 2 } + 13 x + 1$ .

Accepted Answer

The answer of Find all rational zeros of the polynomial...

Question 5

Find all zeros of the polynomial $P ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 3 x - 2$ .

Accepted Answer

@#LAT-DLM& . The zer

Question 6

Use synthetic division and the remainder theorem to evaluate $p ( - 0.1 )$ for $p ( x ) = 2 x ^ { 4 } + x ^ { 3 } - 10 x ^ { 2 } - 1$ .

Accepted Answer

None

Question 7

Find the quotient and remainder of $\frac { 3 x ^ { 5 } - 2 x ^ { 4 } - 5 } { 6 x ^ { 2 } - x + 4 }$ using long division.

Accepted Answer

@#LAT-DLM& . The quo

Question 8

Find all real and complex zeros of $F ( x ) = x ^ { 3 } + x ^ { 2 } - 2$

Accepted Answer

The answer of Find all real and complex zeros of...

Question 9

Find the minimum value of a parabola with $x$ - intercepts of $( 3,0 )$ and $( 12,0 )$ and $y$ - intercept $( 0,18 )$ .

Accepted Answer

Since @#LAT-DLM& , @#LAT-DLM&

Question 10

Find all zeros of the polynomial $P ( x ) = x ^ { 4 } + x ^ { 2 } - 6 x + 4$ .

Accepted Answer

@#LAT-DLM& . The zer

Question 11

If a ball is thrown directly upward with a velocity of 64 ft/s, its height (in feet) after t seconds is given
by $y = 64 t - 16 t ^ { 2 }$ . What is the maximum height attained by the ball?

Accepted Answer

The answer of If a ball is thrown directly upward...

Question 12

A quadratic function is given.
(a) Express the quadratic function in standard form.
(b) Find its vertex and its x- and y-intercept(s).
$f ( x ) = - 3 x ^ { 2 } + 12 x - 11$

Accepted Answer

(a) @#LAT-DLM&
(b) vertex:  (2

Question 13

Find the intercepts and asymptotes, and then graph the rational function $\gamma ( x ) = \frac { 3 x ^ { 2 } + 9 } { x ^ { 2 } - x - 2 }$ .

Accepted Answer

@#LAT-DLM& intercept is @#LAT-DLM& . The function

Question 14

Factor the polynomial $p ( x ) = 256 x ^ { 4 } - 25$ completely and find all its zeros. State the multiplicity of each zero.

Accepted Answer

@#LAT-DLM& . Zeros are @#LAT-DLM& , @#LAT-DLM& , @#LAT-DLM&

Question 15

Find the quotient and remainder using long division. $\frac { x ^ { 6 } + x ^ { 2 } + 4 x ^ { 4 } + 4 } { x ^ { 2 } + 4 }$

Accepted Answer

quotient:

Question 16

Find the remainder if $p ( x ) = 6 x ^ { 342 } + 18 x ^ { 63 } - 5 x ^ { 2 } - 20$ is divided by $( x - 1 )$ .

Accepted Answer

If @#LAT-DLM& is div

Question 17

Find the maximum or the minimum value of the function $f ( x ) = \frac { x } { 4 } \left( 1 - \frac { x } { 4 } \right) + \frac { 5 } { 12 }$ .

Accepted Answer

The answer of Find the maximum or the minimum value...

Question 18

Graph the rational function $y = \frac { x ^ { 3 } - 2 x ^ { 2 } - 5 x + 6 } { x ^ { 2 } + x - 2 }$ by factoring the numerator and denominator.

Accepted Answer

The answer of Graph the rational function \(y = \frac...

Question 19

Graph the family of polynomials $P ( x ) = ( x - 2 ) ^ { c }$ in the same viewing rectangle, using the values $c = 1$ , $3$ , $5$ , and $7$ . Explain how changing the value of $C$ affects the graph.

Accepted Answer

The larger the value

Question 20

Find the quotient and remainder of $\frac { x ^ { 2 } + 2 x - 16 } { x + 6 }$ using long division.

Accepted Answer

@#LAT-DLM& . The quo

Find the quotient and remainder of $\frac { x ^ { 5 } - 3 x ^ { 2 } + 2 x - 2 } { x + 1 }$ .

Use synthetic division and the remainder theorem to evaluate $p ( 3 )$ for $p ( x ) = 2 x ^ { 3 } - 2 x ^ { 2 } + 11 x - 100$ .

Find the maximum or minimum value of the function $f ( x ) = x ^ { 2 } - 4 x + 2$ .

Find all rational zeros of the polynomial $p ( x ) = 15 x ^ { 3 } - 29 x ^ { 2 } + 13 x + 1$ .

Find all zeros of the polynomial $P ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 3 x - 2$ .

Use synthetic division and the remainder theorem to evaluate $p ( - 0.1 )$ for $p ( x ) = 2 x ^ { 4 } + x ^ { 3 } - 10 x ^ { 2 } - 1$ .

Find the quotient and remainder of $\frac { 3 x ^ { 5 } - 2 x ^ { 4 } - 5 } { 6 x ^ { 2 } - x + 4 }$ using long division.

Find all real and complex zeros of $F ( x ) = x ^ { 3 } + x ^ { 2 } - 2$

Find the minimum value of a parabola with $x$ - intercepts of $( 3,0 )$ and $( 12,0 )$ and $y$ - intercept $( 0,18 )$ .

Find all zeros of the polynomial $P ( x ) = x ^ { 4 } + x ^ { 2 } - 6 x + 4$ .

If a ball is thrown directly upward with a velocity of 64 ft/s, its height (in feet) after t seconds is given by $y = 64 t - 16 t ^ { 2 }$ . What is the maximum height attained by the ball?

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its x- and y-intercept(s). $f ( x ) = - 3 x ^ { 2 } + 12 x - 11$

Find the intercepts and asymptotes, and then graph the rational function $\gamma ( x ) = \frac { 3 x ^ { 2 } + 9 } { x ^ { 2 } - x - 2 }$ .

Factor the polynomial $p ( x ) = 256 x ^ { 4 } - 25$ completely and find all its zeros. State the multiplicity of each zero.

Find the quotient and remainder using long division. $\frac { x ^ { 6 } + x ^ { 2 } + 4 x ^ { 4 } + 4 } { x ^ { 2 } + 4 }$

Find the remainder if $p ( x ) = 6 x ^ { 342 } + 18 x ^ { 63 } - 5 x ^ { 2 } - 20$ is divided by $( x - 1 )$ .

Find the maximum or the minimum value of the function $f ( x ) = \frac { x } { 4 } \left( 1 - \frac { x } { 4 } \right) + \frac { 5 } { 12 }$ .

Graph the rational function $y = \frac { x ^ { 3 } - 2 x ^ { 2 } - 5 x + 6 } { x ^ { 2 } + x - 2 }$ by factoring the numerator and denominator.

Graph the family of polynomials $P ( x ) = ( x - 2 ) ^ { c }$ in the same viewing rectangle, using the values $c = 1$ , $3$ , $5$ , and $7$ . Explain how changing the value of $C$ affects the graph.

Find the quotient and remainder of $\frac { x ^ { 2 } + 2 x - 16 } { x + 6 }$ using long division.

Fundamentals

Functions

Exponential and Logarithmic Functions

Trigonometric Functions: Unit Circle Approach

Trigonometric Functions: Right Triangle Approach

Analytic Trigonometry

Polar Coordinates and Parametric Equations

Vectors in Two and Three Dimensions

Systems of Equations and Inequalities

Conic Sections

Sequences and Series

Limits: a Preview of Calculus

Filters

Exam 3: Polynomial and Rational Functions

Find the quotient and remainder of x5−3x2+2x−2x+1\frac { x ^ { 5 } - 3 x ^ { 2 } + 2 x - 2 } { x + 1 }x+1x5−3x2+2x−2​ .

Use synthetic division and the remainder theorem to evaluate p(3)p ( 3 )p(3) for p(x)=2x3−2x2+11x−100p ( x ) = 2 x ^ { 3 } - 2 x ^ { 2 } + 11 x - 100p(x)=2x3−2x2+11x−100 .

Find the maximum or minimum value of the function f(x)=x2−4x+2f ( x ) = x ^ { 2 } - 4 x + 2f(x)=x2−4x+2 .

Find all rational zeros of the polynomial p(x)=15x3−29x2+13x+1p ( x ) = 15 x ^ { 3 } - 29 x ^ { 2 } + 13 x + 1p(x)=15x3−29x2+13x+1 .

Find all zeros of the polynomial P(x)=x3−3x2+3x−2P ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 3 x - 2P(x)=x3−3x2+3x−2 .

Use synthetic division and the remainder theorem to evaluate p(−0.1)p ( - 0.1 )p(−0.1) for p(x)=2x4+x3−10x2−1p ( x ) = 2 x ^ { 4 } + x ^ { 3 } - 10 x ^ { 2 } - 1p(x)=2x4+x3−10x2−1 .

Find the quotient and remainder of 3x5−2x4−56x2−x+4\frac { 3 x ^ { 5 } - 2 x ^ { 4 } - 5 } { 6 x ^ { 2 } - x + 4 }6x2−x+43x5−2x4−5​ using long division.

Find all real and complex zeros of F(x)=x3+x2−2F ( x ) = x ^ { 3 } + x ^ { 2 } - 2F(x)=x3+x2−2

Find the minimum value of a parabola with xxx - intercepts of (3,0)( 3,0 )(3,0) and (12,0)( 12,0 )(12,0) and yyy - intercept (0,18)( 0,18 )(0,18) .

Find all zeros of the polynomial P(x)=x4+x2−6x+4P ( x ) = x ^ { 4 } + x ^ { 2 } - 6 x + 4P(x)=x4+x2−6x+4 .

If a ball is thrown directly upward with a velocity of 64 ft/s, its height (in feet) after t seconds is given by y=64t−16t2y = 64 t - 16 t ^ { 2 }y=64t−16t2 . What is the maximum height attained by the ball?

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its x- and y-intercept(s). f(x)=−3x2+12x−11f ( x ) = - 3 x ^ { 2 } + 12 x - 11f(x)=−3x2+12x−11

Find the intercepts and asymptotes, and then graph the rational function γ(x)=3x2+9x2−x−2\gamma ( x ) = \frac { 3 x ^ { 2 } + 9 } { x ^ { 2 } - x - 2 }γ(x)=x2−x−23x2+9​ .

Factor the polynomial p(x)=256x4−25p ( x ) = 256 x ^ { 4 } - 25p(x)=256x4−25 completely and find all its zeros. State the multiplicity of each zero.

Find the quotient and remainder using long division. x6+x2+4x4+4x2+4\frac { x ^ { 6 } + x ^ { 2 } + 4 x ^ { 4 } + 4 } { x ^ { 2 } + 4 }x2+4x6+x2+4x4+4​

Find the remainder if p(x)=6x342+18x63−5x2−20p ( x ) = 6 x ^ { 342 } + 18 x ^ { 63 } - 5 x ^ { 2 } - 20p(x)=6x342+18x63−5x2−20 is divided by (x−1)( x - 1 )(x−1) .

Find the maximum or the minimum value of the function f(x)=x4(1−x4)+512f ( x ) = \frac { x } { 4 } \left( 1 - \frac { x } { 4 } \right) + \frac { 5 } { 12 }f(x)=4x​(1−4x​)+125​ .

Graph the rational function y=x3−2x2−5x+6x2+x−2y = \frac { x ^ { 3 } - 2 x ^ { 2 } - 5 x + 6 } { x ^ { 2 } + x - 2 }y=x2+x−2x3−2x2−5x+6​ by factoring the numerator and denominator.

Graph the family of polynomials P(x)=(x−2)cP ( x ) = ( x - 2 ) ^ { c }P(x)=(x−2)c in the same viewing rectangle, using the values c=1c = 1c=1 , 333 , 555 , and 777 . Explain how changing the value of CCC affects the graph.

Find the quotient and remainder of x2+2x−16x+6\frac { x ^ { 2 } + 2 x - 16 } { x + 6 }x+6x2+2x−16​ using long division.

Fundamentals

Functions

Exponential and Logarithmic Functions

Trigonometric Functions: Unit Circle Approach

Trigonometric Functions: Right Triangle Approach

Analytic Trigonometry

Polar Coordinates and Parametric Equations

Vectors in Two and Three Dimensions

Systems of Equations and Inequalities

Conic Sections

Sequences and Series

Limits: a Preview of Calculus

Filters

Find the quotient and remainder of $\frac { x ^ { 5 } - 3 x ^ { 2 } + 2 x - 2 } { x + 1 }$ .

Use synthetic division and the remainder theorem to evaluate $p ( 3 )$ for $p ( x ) = 2 x ^ { 3 } - 2 x ^ { 2 } + 11 x - 100$ .

Find the maximum or minimum value of the function $f ( x ) = x ^ { 2 } - 4 x + 2$ .

Find all rational zeros of the polynomial $p ( x ) = 15 x ^ { 3 } - 29 x ^ { 2 } + 13 x + 1$ .

Find all zeros of the polynomial $P ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 3 x - 2$ .

Use synthetic division and the remainder theorem to evaluate $p ( - 0.1 )$ for $p ( x ) = 2 x ^ { 4 } + x ^ { 3 } - 10 x ^ { 2 } - 1$ .

Find the quotient and remainder of $\frac { 3 x ^ { 5 } - 2 x ^ { 4 } - 5 } { 6 x ^ { 2 } - x + 4 }$ using long division.

Find all real and complex zeros of $F ( x ) = x ^ { 3 } + x ^ { 2 } - 2$

Find the minimum value of a parabola with $x$ - intercepts of $( 3,0 )$ and $( 12,0 )$ and $y$ - intercept $( 0,18 )$ .

Find all zeros of the polynomial $P ( x ) = x ^ { 4 } + x ^ { 2 } - 6 x + 4$ .

If a ball is thrown directly upward with a velocity of 64 ft/s, its height (in feet) after t seconds is given by $y = 64 t - 16 t ^ { 2 }$ . What is the maximum height attained by the ball?

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its x- and y-intercept(s). $f ( x ) = - 3 x ^ { 2 } + 12 x - 11$

Find the intercepts and asymptotes, and then graph the rational function $\gamma ( x ) = \frac { 3 x ^ { 2 } + 9 } { x ^ { 2 } - x - 2 }$ .

Factor the polynomial $p ( x ) = 256 x ^ { 4 } - 25$ completely and find all its zeros. State the multiplicity of each zero.

Find the quotient and remainder using long division. $\frac { x ^ { 6 } + x ^ { 2 } + 4 x ^ { 4 } + 4 } { x ^ { 2 } + 4 }$

Find the remainder if $p ( x ) = 6 x ^ { 342 } + 18 x ^ { 63 } - 5 x ^ { 2 } - 20$ is divided by $( x - 1 )$ .

Find the maximum or the minimum value of the function $f ( x ) = \frac { x } { 4 } \left( 1 - \frac { x } { 4 } \right) + \frac { 5 } { 12 }$ .

Graph the rational function $y = \frac { x ^ { 3 } - 2 x ^ { 2 } - 5 x + 6 } { x ^ { 2 } + x - 2 }$ by factoring the numerator and denominator.

Graph the family of polynomials $P ( x ) = ( x - 2 ) ^ { c }$ in the same viewing rectangle, using the values $c = 1$ , $3$ , $5$ , and $7$ . Explain how changing the value of $C$ affects the graph.

Find the quotient and remainder of $\frac { x ^ { 2 } + 2 x - 16 } { x + 6 }$ using long division.