Question 1

Determine the maximum number of turning points of f.
-$$f ( x ) = x ^ { 2 } ( x + 2 )$$
Answer: (a) For large values of $| x |$, the graph of $f ( x )$ will resemble the graph of $y = x ^ { 3 }$.
(b) $y$-intercept: $( 0,0 )$, $x$-intercepts: $( 0,0 )$ and $( - 2,0 )$
(c) The graph of $f$ crosses the $x$-axis at $( - 2,0 )$ and touches the $x$-axis at $( 0,0 )$.
(e) Local minimum at $( 0,0 )$, Local maximum at $( - 1.33,1.19 )$
(f)

(g) Domain of f: all real numbers; range of f: all real numbers
(h) f is increasing on (-, -1.33) and (0, ); f is decreasing on (-1.33, 0)

Accepted Answer

The answer of Determine the maximum number of turning points...

Question 2

Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval.
-f(x) = x5 - 10x4 + 42x3 -124 x2 + 297x - 306 ; zero: 3i

Accepted Answer

A)  2, -3i, 4 - i, 4 + i 
B)  -2, -3i, -4 - i, -4 + i 
C)  2, -3i, -4 - i, -4 + i 
D)  -2, -3i, 4 - i, 4 + i

Question 3

Find the x- and y-intercepts of f.
-$$f ( x ) = ( x + 6 ) ^ { 2 }$$

Accepted Answer

A)  x-intercept: -6; y-intercept: 36 
B)  x-intercept: -6; y-intercept: 0 
C)  x-intercept: 6; y-intercept: 36 
D)  x-intercept: 6; y-intercept: 0

Question 4

Find the x- and y-intercepts of f.
-$$f ( x ) = 2 x ^ { 4 } ( x - 5 ) ^ { 5 }$$

Accepted Answer

A)  x-intercepts: 0, -5; y-intercept: 2 
B)  x-intercepts: 0, 5; y-intercept: 2 
C)  x-intercepts: 0, -5; y-intercept: 0 
D)  x-intercepts: 0, 5; y-intercept: 0

Question 5

$$f ( x ) = 5 \left( x ^ { 2 } + 6 \right) \left( x ^ { 2 } + 4 \right) ^ { 2 }$$

Accepted Answer

A)  $\sqrt { 6 }$, multiplicity 1 , crosses $x$-axis; $- \sqrt { 6 }$, multiplicity 1 , crosses $x$-axis; 2 , multiplicity 2 , touches $x$-axis; $- 2$, multiplicity 2 , touches $x$-axis 
B)  -6, multiplicity 1 , touches $x$-axis; $- 4$, multiplicity 2, crosses $x$-axis 
C)  -6, multiplicity 1 , crosses $x$-axis; $- 4$, multiplicity 2 , touches $x$-axis 
D)  No real zeros

Question 6

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not.
-f(x) = 19x5 + 9x4 + 2

Accepted Answer

A)  Yes; degree 9 
B)  No; the last term has no variable 
C)  Yes; degree 10 
D)  Yes; degree 5

Question 7

Find the x- and y-intercepts of f.
-$$f ( x ) = - x ^ { 2 } ( x + 6 ) \left( x ^ { 2 } + 1 \right)$$ $$( x ) = - x ^ { 2 } ( x + 6 ) \left( x ^ { 2 } + 1 \right)$$

Accepted Answer

A)  x-intercepts: -6, -1, 0; y-intercept: 6 
B)  x-intercepts: -6, -1, 0; y-intercept: -6 
C)  x-intercepts: -6, -1, 0, 1; y-intercept: 0 
D)  x-intercepts: -6, 0; y-intercept: 0

Question 8

Solve the equation in the real number system.
-$$x ^ { 4 } - 3 x ^ { 3 } + 5 x ^ { 2 } - x - 10 = 0$$

Accepted Answer

A)  {-1, 2} 
B)  {-1, -2} 
C)  {1, 2} 
D)  {-2, 1}

Question 9

Solve the equation in the real number system.
-$$x ^ { 3 } + 3 x ^ { 2 } - 10 x - 24 = 0$$

Accepted Answer

A)  {-4, -2} 
B)  {2, 4} 
C)  {-3, 2, 4} 
D)  {-4, -2, 3}

Question 10

Find the x- and y-intercepts of f.
-f(x) = (x - 2)(x - 5)

Accepted Answer

A)  x-intercepts: -2, -5; y-intercept: 10 
B)  x-intercepts: 2, 5; y-intercept: -7 
C)  x-intercepts: 2, 5; y-intercept: 10 
D)  x-intercepts: -2, -5; y-intercept: -7

Question 11

Accepted Answer

A) 
 
B) 
 
C) 
 
D)

Question 12

Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval.
-f(x) = 3x3 - 6x + 2; [1, 2]

Accepted Answer

A)  f(1) = -1 and f(2) = -14; no 
B)  f(1) = -1 and f(2) = 14; yes 
C)  f(1) = 1 and f(2) = -14; yes 
D)  f(1) = 1 and f(2) = 14; no

Question 13

The equation has a solution r in the interval indicated. Approximate this solution correct to two decimal places. -x³ - 8x - 3 = 0; -1 ≤ r ≤ 0

Accepted Answer

The answer of The equation has a solution r in...

Question 14

Determine the maximum number of turning points of f.
-f(x) = 8x - x3

Accepted Answer

A)  1 
B)  3 
C)  2 
D)  4

Question 15

The equation has a solution r in the interval indicated. Approximate this solution correct to two decimal places. -x⁴ - x³ - 7x² + 5x + 10 = 0; 2 < r ≤ 3

Accepted Answer

The answer of The equation has a solution r in...

Question 16

Find the domain of the rational function.
-$$g ( x ) = \frac { 6 x } { x - 3 }$$

Accepted Answer

A)  $\{ x \mid x \neq - 3 \}$ 
B)  $\{ x \mid x \neq 3 \}$ 
C)  $\{ x \mid x \neq 0 \}$ 
D)  all real numbers

Exam 3: Polynomial and Rational Functions

Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval. -f(x) = x⁵ - 10x⁴ + 42x³ -124 x² + 297x - 306 ; zero: 3i

Find the x- and y-intercepts of f. - $f ( x ) = ( x + 6 ) ^ { 2 }$

Find the x- and y-intercepts of f. - $f ( x ) = 2 x ^ { 4 } ( x - 5 ) ^ { 5 }$

$f ( x ) = 5 \left( x ^ { 2 } + 6 \right) \left( x ^ { 2 } + 4 \right) ^ { 2 }$

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. -f(x) = 19x⁵ + 9x⁴ + 2

Find the x- and y-intercepts of f. - $f ( x ) = - x ^ { 2 } ( x + 6 ) \left( x ^ { 2 } + 1 \right)$ $( x ) = - x ^ { 2 } ( x + 6 ) \left( x ^ { 2 } + 1 \right)$

Solve the equation in the real number system. - $x ^ { 4 } - 3 x ^ { 3 } + 5 x ^ { 2 } - x - 10 = 0$

Solve the equation in the real number system. - $x ^ { 3 } + 3 x ^ { 2 } - 10 x - 24 = 0$

Find the x- and y-intercepts of f. -f(x) = (x - 2) (x - 5)

Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval. -f(x) = 3x³ - 6x + 2; [1, 2]

The equation has a solution r in the interval indicated. Approximate this solution correct to two decimal places. -x³ - 8x - 3 = 0; -1 ≤ r ≤ 0

Determine the maximum number of turning points of f. -f(x) = 8x - x³

The equation has a solution r in the interval indicated. Approximate this solution correct to two decimal places. -x⁴ - x³ - 7x² + 5x + 10 = 0; 2 < r ≤ 3

Find the domain of the rational function. - $g ( x ) = \frac { 6 x } { x - 3 }$

Exam 3: Polynomial and Rational Functions

Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval. -f(x) = x5 - 10x4 + 42x3 -124 x2 + 297x - 306 ; zero: 3i

Find the x- and y-intercepts of f. - f(x) =(x+6) 2f ( x ) = ( x + 6 ) ^ { 2 }f(x) =(x+6) 2

Find the x- and y-intercepts of f. - f(x) =2x4(x−5) 5f ( x ) = 2 x ^ { 4 } ( x - 5 ) ^ { 5 }f(x) =2x4(x−5) 5

f(x) =5(x2+6) (x2+4) 2f ( x ) = 5 \left( x ^ { 2 } + 6 \right) \left( x ^ { 2 } + 4 \right) ^ { 2 }f(x) =5(x2+6) (x2+4) 2

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. -f(x) = 19x5 + 9x4 + 2

Find the x- and y-intercepts of f. - f(x) =−x2(x+6) (x2+1) f ( x ) = - x ^ { 2 } ( x + 6 ) \left( x ^ { 2 } + 1 \right) f(x) =−x2(x+6) (x2+1) (x) =−x2(x+6) (x2+1) ( x ) = - x ^ { 2 } ( x + 6 ) \left( x ^ { 2 } + 1 \right) (x) =−x2(x+6) (x2+1)

Solve the equation in the real number system. - x4−3x3+5x2−x−10=0x ^ { 4 } - 3 x ^ { 3 } + 5 x ^ { 2 } - x - 10 = 0x4−3x3+5x2−x−10=0

Solve the equation in the real number system. - x3+3x2−10x−24=0x ^ { 3 } + 3 x ^ { 2 } - 10 x - 24 = 0x3+3x2−10x−24=0

Find the x- and y-intercepts of f. -f(x) = (x - 2) (x - 5)

Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval. -f(x) = 3x3 - 6x + 2; [1, 2]

The equation has a solution r in the interval indicated. Approximate this solution correct to two decimal places. -x3 - 8x - 3 = 0; -1 ≤ r ≤ 0

Determine the maximum number of turning points of f. -f(x) = 8x - x3

The equation has a solution r in the interval indicated. Approximate this solution correct to two decimal places. -x4 - x3 - 7x2 + 5x + 10 = 0; 2 < r ≤ 3

Find the domain of the rational function. - g(x) =6xx−3g ( x ) = \frac { 6 x } { x - 3 }g(x) =x−36x​

Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval. -f(x) = x⁵ - 10x⁴ + 42x³ -124 x² + 297x - 306 ; zero: 3i

Find the x- and y-intercepts of f. - $f ( x ) = ( x + 6 ) ^ { 2 }$

Find the x- and y-intercepts of f. - $f ( x ) = 2 x ^ { 4 } ( x - 5 ) ^ { 5 }$

$f ( x ) = 5 \left( x ^ { 2 } + 6 \right) \left( x ^ { 2 } + 4 \right) ^ { 2 }$

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. -f(x) = 19x⁵ + 9x⁴ + 2

Find the x- and y-intercepts of f. - $f ( x ) = - x ^ { 2 } ( x + 6 ) \left( x ^ { 2 } + 1 \right)$ $( x ) = - x ^ { 2 } ( x + 6 ) \left( x ^ { 2 } + 1 \right)$

Solve the equation in the real number system. - $x ^ { 4 } - 3 x ^ { 3 } + 5 x ^ { 2 } - x - 10 = 0$

Solve the equation in the real number system. - $x ^ { 3 } + 3 x ^ { 2 } - 10 x - 24 = 0$

Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval. -f(x) = 3x³ - 6x + 2; [1, 2]

The equation has a solution r in the interval indicated. Approximate this solution correct to two decimal places. -x³ - 8x - 3 = 0; -1 ≤ r ≤ 0

Determine the maximum number of turning points of f. -f(x) = 8x - x³

The equation has a solution r in the interval indicated. Approximate this solution correct to two decimal places. -x⁴ - x³ - 7x² + 5x + 10 = 0; 2 < r ≤ 3

Find the domain of the rational function. - $g ( x ) = \frac { 6 x } { x - 3 }$