algebra 2a - unit 3: polynomial functions lesson 9-14

what is the pythagorean triple if x = 8 and y = 3?

48, 55, and 73

lesson 9

factoring and solving polynomials

lesson 10

polynomial identities

what are the zeros of the polynomial function p(x) = 16x^4 - 8x^2 + 1?

x = -1/2 and x = 1/2

the polynomial function f(n) = 15n^4− 45n^3 + 12n^2 − 36n can be factored as f(n) = 3n(5n^3 + 4)(n - 3). what are all of the zeros of the polynomial function?

n = 0, n = 3, n = - 2i√5 / 5, and n = 2i√5 / 5

how can (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 be derived from the polynomial identity (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3?

substitute −b for b on both sides of (a − b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

consider the graph of the cubic function f(x). https://cdstools.flipswitch.com/asset/media/1236460 which statements are true about the function represented by the graph? select all that apply.

the function has three x-intercepts at (-2, 0), (2, 0) and (3, 0), and a y-intercept at (0, 12)

which polynomial expression is equal to m(m + 2)(m^2 - 2m + 4)?

m^4 + 8m

which statements are true about the function f(x) = −(x + 5)(x − 1)(x − 2)? select all that apply.

the y-intercept is (0, -10) the x-intercepts are (−5, 0), (1, 0), and (2, 0) the domain is all real numbers, and the range is all real numbers.

what is the factored form of the polynomial expression b^4 - 81?

(b + 3)(b - 3)(b^2 + 9)

which expression correctly shows the factored form of b^4 - 81c^2?

(b^2 - 9c)(b^2 + 9c)

read the information and then follow the instructions. given: (x^2 + 1)^2 prove: (x^2 + 1)^2 = (x^2 - 1)^2 + (2x)^2 for steps 4 through 9, match each numbered statement with the correct reason.

0.67 of 1 4. distributive property 5. combine like terms 6. substitute 7. commutative property of addition 8. power of a product rule 9. rewrite by using the perfect square trinomial pattern.

use the information to complete the proof. given: b(5 - b)(5 + b) prove: b(5 - b)(5 + b) = 25b - b^3 which statement correctly fills in the blank for statement 3 to complete the proof?

5b(5 + b) − b^2(5 + b)

examine the following table, which contains some of the points of a cubic function, f(x). x: -2, -1.5, -1... f(x): -8.75, -3, 0 which statements can be true about the function represented in the table? select all that apply.

as x approaches −∞, f(x) approaches −∞, and as x approaches ∞, f(x) approaches ∞. the function has x-intercepts of (−1, 0), (0.5, 0), and (1.5, 0), and a y-intercept of (0, 0.75). the function has a relative maximum over the interval (−1, 0) and a relative minimum over the interval (0.5, 1.5)

lesson 11

key features from graphs: cubic functions

consider the function f(x) represented by the graph. https://cdstools.flipswitch.com/asset/media/1236473 match each phrase with the part that correctly completes the related sentence.

the domain and range are : all real numbers the y-intercept is : (0, 4) the zeros are : x = -2.5, x = -0.5, and x = 3.1 the function is positive over : (−∞, −2.5) and (-0.5, 3.1) the function is negative over : (-2.5, -0.5) and (3.1, ∞)

which expression is equivalent to 9x^4 + 6x^2 + 1?

(3x^2 + 1)(3x^2 + 1)

which expression shows the factored form of the expression 64m^3 + 125?

(4m + 5)(16m^2 - 20m + 25)

erdem is going to construct a rectangular paper box. he will make the box x inches wide. the length of the box will be 2 inches more than its width, and the height of the box will be 4 inches more than its width. considering the possible values for the volume of the box, which cubic inequality can help erdem find the possible values of the box's width, x?

0 < x(x + 2)(x + 4)

read the information and then follow the instructions. given: (x^3 + 4)^2 prove: (x^3 + 4)^2 = (x^2 − 4)^2 + (4x)^2​ for steps 4 through 9, match each numbered statement with the correct reason.

0.67 of 1 4. distributive property 5. combine like terms 6. substitute 7. commutative property of addition 8. power of a product rule 9. rewrite by using the perfect square trinomial pattern.

which expression correctly shows 2x^4 - 54x factored using the difference of cubes method?

2x(x - 3)(x^2 + 3x + 9)

isaiah constructed a rectangular open paper box. he first cut squares from the corners of a sheet of paper that measured 19 inches by 25 inches, and then he folded up the sides. the equation that represents the volume of the open paper box is v(x) = x(19 - 2x)(25 - 2x), where x represents the length of one side of each square. if the volume of the open box is 756.19675 in.3, what is the length of one side of each square?

3.57 inches

if the sides of a right triangle are 140, 171, and 221 units, what are the values of x and y?

5, 14

which expression correctly shows the factors for 20x^4 - 25x^2 + 60x^2 - 75x?

5x(x^2 + 3)(4x - 5)

if the sides of a right triangle are 57, 176, and 185, what are the values of x and y?

8, 11

which statements are true about function f(x) = (x - 3)(x + 1)(x - 2)? select all that apply.

as x approaches -inf, (x) approaches -inf, and as x approaches inf, f(x) approaches inf the function has a relative maximum between the x-values -1 and 2, and it has a relative minimum between the x-values 2 and 3 the function is positive over the intervals (-1, 2) and (3, inf), and the function is negative over the intervals (-inf, -1) and (2, 3)

consider the graph of f(x). https://cdstools.flipswitch.com/asset/media/1236474 which statement is true about the function represented by the graph?

as x approaches negative infinity, f(x) approaches infinity, and as x approaches infinity, f(x) approaches negative infinity.

consider the graph of f(x) https://cdstools.flipswitch.com/asset/media/1236471 which statement is true about the function represented by the graph?

as x approaches negative infinity, f(x) approaches negative infinity, and as x approaches infinity, f(x) approaches infinity.

which statements are true about the function f(x) = x^3 + 2x^2 − 9x − 18? select all that apply.

as x approaches −∞, f(x) approaches −∞, and as x approaches ∞, f(x) approaches ∞. the function has a relative maximum between the x-values −3 and −2, and it has a relative minimum between the x-values −2 and 3. the function is positive over the intervals (−3, −2) and (3, ∞), and the function is negative over the intervals (−∞, −3) and (−2, 3).

which statements are true about the function f(x) = −(x − 1)^2(x + 3)? select all that apply.

as x approaches −∞, f(x) approaches ∞, and as x approaches ∞, f(x) approaches −∞. the function has a relative minimum between x = −3 and x = 1, and it has a relative maximum at x = 1. the function is positive over the interval (−∞, −3), and the function is negative over the interval (3, ∞), except for when x = 1.

which statements are true about the function f(x) = −(x + 5)(x − 1)(x − 2)? select all that apply.

as x approaches −∞, f(x) approaches ∞, and as x approaches ∞, f(x) approaches −∞. the function is positive over the intervals (−∞, −5) and (1, 2), and the function is negative over the intervals (−5, 1) and (2, ∞) the function has a relative minimum over the interval (−5, 1), and it has a relative maximum over the interval (1, 2)

examine the following table, which contains some of the points of a cubic function, f(x). x: -2, -1, 0, 1, 2, 3, 4 f(x): 27, 8, 1, 0, -1, -8, -27 which graph represents the same function?

https://cdstools.flipswitch.com/asset/media/1236649

examine the following table, which contains some of the points of a cubic function, f(x). x: -1, 0, 1, 2, 3, 4 f(x): -16, -3, 0, -1, 0, 9 which graph represents the same function?

https://cdstools.flipswitch.com/asset/media/1236657

examine the following table, which contains some of the points of a cubic function, f(x). x: -2, -1.5, -1, 0.5... f(x): -8.75, -3, 0, 1 which graph represents the same funtion?

https://cdstools.flipswitch.com/asset/media/1236690

examine the following table, which contains some of the points of a cubic function, f(x). x: -3, -2,-1, 0, 1, 2 f(x): 5, 0, 3, 8, 9, 0 which graph represents the same function?

https://cdstools.flipswitch.com/asset/media/1236694

examine the following table, which contains some of the points of a cubic function, f(x). x: -3, -2, -1, 0, 1, 2, 3 f(x): -8, -1, 0, 1, 8, 27, 64 which graph represents the same function?

https://cdstools.flipswitch.com/asset/media/1236699

which graph represents the function that has the rule f(x) = (x − 3)(x + 1)(x − 2)?

https://cdstools.flipswitch.com/asset/media/1236920

which graph represents the function that has the rule f(x) = −(x − 4)^3?

https://cdstools.flipswitch.com/asset/media/1236924

which graph represents the function that has the rule f(x) = x^3 + 2x^2 − 9x − 18?

https://cdstools.flipswitch.com/asset/media/1236928

which graph represents the function that has the rule f(x) = −(x − 1)^2(x + 3)?

https://cdstools.flipswitch.com/asset/media/1236932

which graph represents the function that has the rule f(x) = −(x + 5)(x − 1)(x − 2)?

https://cdstools.flipswitch.com/asset/media/1236940

which graph represents the function that has the rule f(x) = x^3 − x^2 − 4x + 4?

https://cdstools.flipswitch.com/asset/media/1236944

isaiah constructed a rectangular open paper box. he first cut squares from the corners of a sheet of paper that measured 19 inches by 25 inches, and then he folded up the sides. which graph has the most appropriate labels and scales to graph the function?

https://cdstools.flipswitch.com/asset/media/1237003

lesson 12

key features from tables: cubic functions

select the solution(s) to the polynomial equation 0 = m^3 + 6m^2 + 9m

m = 0 and m = -3

the polynomial p(x) = 2x^3 - 5x^2 - 42x can be factored as p(x) = x(x - 6)(2x + 7). what are all the zeros of the polynomial function?

m = 0, m = 6, and m = -7/2

how can (a + 3)^2 = a^2 + 6a + 9 be derived from the polynomial identity (a - 3)^2 = a^2 - 6a + 9?

substitute -a for a on both sides

which statements are true about the function f(x) = (x - 3)(x + 1)(x - 2)? select all that apply.

the domain is all real numbers, and the range is all real numbers. the x-intercepts are (-1, 0), (2, 0), and (3, 0) the y-intercept is (0, 6)

consider the graph of f(x). https://cdstools.flipswitch.com/asset/media/1236461 which statements are true about the function represented by the graph? select all that apply.

the function decreases over the interval (−0.5, 2.5) the function increases over the intervals (−∞, −0.5) and (2.5, ∞)

which statements are true about the function f(x) = -(x - 4)^3? select all that apply.

the function does not have a relative maximum or relative minimum. as x approaches −∞, f(x) approaches ∞, and as x approaches ∞, f(x) approaches −∞. the function is positive over the interval (−∞, 4), and the function is negative over the interval (4, ∞).

consider the graph of f(x). https://cdstools.flipswitch.com/asset/media/1236461 which statement is true about the function represented by the graph?

the function has a relative maximum of 13.1 and a relative minimum of −1.1

examine the following table, which contains some of the points of a cubic function, f(x) x: -2, -1.5, -1, -0.5..... f(x): 27, 8, 0, -1.875... which statement can be true about the function represented in the table?

the function has a relative minimum over the interval (−0.5, 0.5) and a relative maximum over the interval (1, 2).

examine the following table, which contains some of the points of a cubic function, f(x). x: -3, -2, -1, 0, 1, 2, 3, 4 f(x): 5, 0, 3, 8, 9, 0, -25, -72 which statement can be true about the function represented in the table?

the function has a relative minimum over the interval (−3, −1) and a relative maximum over the interval (0, 2)

examine the following table, which contains some of the points of a cubic function, f(x). x: -3, -2, -1... f(x): -8, -1, 0 which statements can be true about the function represented in the table? select all that apply.

the function has no relative maximum and no relative minimum. the function has an x-intercept of (−1, 0) and a y-intercept of (0, 1). as x approaches −∞, f(x) approaches −∞, and as x approaches ∞, f(x) approaches ∞.

examine the following table, which contains some of the points of a cubic function, f(x). x: -3, -2, -1, 0... f(x): -96, -45, -16, -3 which statements can be true about the function represented in the table? select all that apply.

the function has x-intercepts of (1, 0) and (3, 0), and a y-intercept of (0, −3). the function has a relative maximum between the x-values of 0 and 2, and a relative minimum between the x-values of 1 and 3. as x approaches −∞, f(x) approaches −∞, and as x approaches ∞, f(x) approaches ∞.

consider the graph of f(x) https://cdstools.flipswitch.com/asset/media/1236475 which statement is true about the function represented by the graph?

the function is increasing over (−1.6, 1.6) and is decreasing over (−∞, −1.6) and (1.6, ∞).

consider the graph of the cubic function f(x). https://cdstools.flipswitch.com/asset/media/1236460 which statement is true about the function represented by the graph?

the function is positive over the intervals (−2, 2) and (3, ∞) the function is negative over the intervals (−∞, −2) and (2, 3)

robyn wants to construct a rectangular metal box that has a length of x inches, a width of 3 inches less than its length, and a height that is 1 inch more than its length. the function that represents the volume of the box for this situation is v = x^3 − 2x^2 − 3x.

the range for the situation involves all y-values greater than 0 the domain for the situation involves all x-values greater than 3 to have a volume of 125.84 cubic inches, the length of the box should be 6 inches.

which statements are true about the function f(x) = -(x - 4)^3? select all that apply.

the x-intercept is (4, 0) the y-intercept is (0, 64) domain: (-inf, inf) range: (-inf, inf)

which statements are true about the function f(x) = x^3 + 2x^2 - 9x - 18? select all that apply.

the y-intercept is (0, -18) the x-intercepts are (-3, 0), (-2, 0), and (3, 0) domain: -inf < x < inf range: -inf < y < inf

which statements are true about the function f(x) = -(x - 1)^2 (x + 3)? select all that apply.

the y-intercept is (0, -3) the x-intercepts are (-3, 0) and (1, 0) domain: (-inf, inf) range: (-inf, inf)

which statements are true about the function f(x) = x^3 − x^2 − 4x + 4? select all that apply.

the y-intercept is (0, 4) the x-intercepts are (-2, 0), (1, 0), and (2, 0) the domain is all real numbers, and the range is all real numbers.

consider the graph of f(x) https://cdstools.flipswitch.com/asset/media/1236474 which statements are true about the function represented by the graph? select all that apply.

the y-intercept is (0, 4) the zeros are x = -2 and x = 1 the range of the function is all real numbers. the domain of the function is all real numbers.

checkpoint; consider the graph of f(x). https://cdstools.flipswitch.com/asset/media/1236472 which statements are true about the function represented by the graph? select all that apply.

the y-intercept is (0, 5) the x-intercept is (-2, 0) the domain of the function is all real numbers. the range of the function is all real numbers.

consider the graph of f(x). https://cdstools.flipswitch.com/asset/media/1236459 which statements are true about the function represented by the graph? select all that apply.

the zero of the function is x = -2 the range of the function is (−∞, ∞) the domain of the function is (−∞, ∞)

examine the following table, which contains some of the points of a cubic function, f(x). x: -3, -2, -1, 0, 1, 2, 3, 4 f(x): 125, 27, 8, 1, 0, -1, -8, -27 which statements can be true about the function represented in the table? select all that apply.

there are no relative maxima or minima of the function. the function has an x-intercept of (1, 0) and a y-intercept of (0, 1) as x approaches −∞, f(x) approaches ∞,∞, and as x approaches ∞, f(x) approaches −∞.

consider the graph of f(x) https://cdstools.flipswitch.com/asset/media/1236472 which statements are true about the function represented by the graph? select all that apply.

there are no relative maxima or relative minima for this function. as x approaches negative infinity, f(x) approaches negative infinity, and as x approaches infinity, f(x) approaches infinity.

isaiah constructed a rectangular open paper box. he first cut squares from the corners of a sheet of paper that measured 19 inches by 25 inches, and then he folded up the sides. which cubic function, v(x), represents the volume of the box, where x represents the side length of each square?

v(x) = x(19 − 2x)(25 − 2x)

if 96, 247, and 265 are the sides of a right triangle, what are the values of x and y?

x = 16, y = 3

what are the solutions to the polynomial equation 27x^3 − 8 = 0?

x = 2/3, x = -1 + i√3 / 3, and x = -1 - i√3 / 3

erdem is going to construct a rectangular paper box. he will make the box x inches wide. the length of the box will be 2 inches more than its width, and the height of the box will be 4 inches more than its width. considering the possible values for the volume of the box, which answers are possible values of the box's width, x inches? select all that apply.

x = 5 x = 8 x = 14

what are the solutions to the polynomial equation x^3 - 125 = 0?

x = 5, x = -5 + 5i√3 / 2, and x = -5 - 5i√3 / 2

which expression correctly shows 27x^3 - 8 factored over the integers using the difference of cubes method?

(3x - 2)(9x^2 + 6x + 4)

fabiana is going to construct a rectangular metal box with a volume of 150 in.3. her box will have a length of x inches, a width of 3 inches less than its length, and a height that is 1 inch more than its length. if the cubic equation is x^3 - 2x^2 - 3x - 150 = 0 represents this situation, what is the length of the box in inches?

6.28 inches

examine the following table, which contains some of the points of a cubic function, f(x). x: -1.5, -1, -0.5 f(x): 4.375, 0, -1.875... which graph represents the same function?

https://cdstools.flipswitch.com/asset/media/1236653

what is the factored form of 4a^4 - 4a^2 + 1?

(2a^2 - 1)^2

which statements are true about the function f(x) = x^3 − x^2 − 4x + 4? select all that apply.

as x approaches −∞, f(x) approaches −∞, and as x approaches ∞, f(x) approaches ∞. the function has a relative maximum over the interval (−2, 1), and it has a relative minimum over the interval (1, 2). the function is positive over the intervals (−2, 1) and (2, ∞), and the function is negative over the intervals (−∞, −2) and (1, 2)

use the information to complete the proof. given: (a + 2)^2 prove: (a + 2)^2 = a^2 + 4a + 4 which statement correctly fills in the blank for statement 3 to complete the proof?

a(a + 2) + 2(a + 2)

use the information to complete the proof. given: (a - b)^3 prove: (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 which statement correctly fills in the blank for statement 5 to complete the proof?

a^3 − 2a^2b + ab^2 − a^2b + 2ab^2 − b^3

which polynomial expression is equal to (a + 3)(a − 3)(a^2 + 9)?

a^4 - 81

lesson 14

cubic equations and inequalities in context

examine the graph. https://cdstools.flipswitch.com/asset/media/1236948 which function is represented by the graph?

x^3 + 4x^2 + 4x + 3

match each polynomial in standard form to its equivalent factored form.

x^3 + 8 : (x + 2)(x^2 - 2x + 4) 2x^4 + 16x : 2x(x + 2)(x^2 - 2x + 4) 8x^3 + 1 : (2x + 1)(4x^2 - 2x + 1)

examine the graph. https://cdstools.flipswitch.com/asset/media/1236936 which function is represented by the graph?

x^3 - 3x^2 - 3x - 4

if 17, 144, and 145 are the sides of a right triangle, what are the values of x and y?

x = 9 and y = 8

fabiana is going to construct a rectangular metal box with a volume of 150 in.3. her box will have a length of x inches, a width of 3 inches less than its length, and a height that is 1 inch more than its length. which cubic equation would help fabiana find the length of the box?

x^3 − 2x^2 − 3x − 150 = 0

lesson 13

key features from rules: cubic functions