Podcast
Questions and Answers
Which expression correctly shows $2x^3y - 2y$ factored completely?
Which expression correctly shows $2x^3y - 2y$ factored completely?
- 2xy(x^2 - 1)
- 2y(x − 1)(x^2 + x + 1) (correct)
- 2y(x^3 - 1)
- 2y(x + 1)(x^2 - x + 1)
Which expression correctly shows the factored form of $n^3 + rac{27}{125}$?
Which expression correctly shows the factored form of $n^3 + rac{27}{125}$?
- (n + 3)(n^2 - 3n + 9)
- (n + rac{1}{5})(n^2 - rac{1}{5}n + rac{1}{25})
- (n + rac{3}{5})(n^2 - rac{3}{5}n + rac{9}{25}) (correct)
- (n + 5)(n^2 - 5n + 25)
What is the factored form of $121x^4 - 9y^2$?
What is the factored form of $121x^4 - 9y^2$?
(11x^2 - 3y)(11x^2 + 3y)
What are the solutions to the polynomial equation $64x^3 + 1 = 0$?
What are the solutions to the polynomial equation $64x^3 + 1 = 0$?
What are all of the zeros of the polynomial function $f(a) = a^4 - 81$?
What are all of the zeros of the polynomial function $f(a) = a^4 - 81$?
Which statement correctly fills in the blank for statement 2 to complete the proof?
Which statement correctly fills in the blank for statement 2 to complete the proof?
Which polynomial expression is equal to $(2 - x)(2 + x)(4 + x^2)$?
Which polynomial expression is equal to $(2 - x)(2 + x)(4 + x^2)$?
Match each numbered statement with the correct reason.
Match each numbered statement with the correct reason.
If 36, 77, and 85 are the sides of a right triangle, what are the values of x and y?
If 36, 77, and 85 are the sides of a right triangle, what are the values of x and y?
Which statements are true about the function represented by the graph of f(x)? (Select all that apply)
Which statements are true about the function represented by the graph of f(x)? (Select all that apply)
Which statements are true about the function represented by the graph of f(x)? (Select all that apply)
Which statements are true about the function represented by the graph of f(x)? (Select all that apply)
Which statements can be true about the function represented in the table? (Select all that apply)
Which statements can be true about the function represented in the table? (Select all that apply)
Which graph represents the same function as given in the table?
Which graph represents the same function as given in the table?
Which statements are true about the function $f(x) = (x - 2)^2(x + 1)$? (Select all that apply)
Which statements are true about the function $f(x) = (x - 2)^2(x + 1)$? (Select all that apply)
Which statements are true about the function $f(x) = (x - 2)^2(x + 1)$? (Select all that apply)
Which statements are true about the function $f(x) = (x - 2)^2(x + 1)$? (Select all that apply)
Which graph represents the function that has the rule $f(x) = (x - 2)^2(x + 1)$?
Which graph represents the function that has the rule $f(x) = (x - 2)^2(x + 1)$?
What cubic equation would help Eren find the length of the box?
What cubic equation would help Eren find the length of the box?
Which cubic inequality can help Eileen find the possible values of the width, x?
Which cubic inequality can help Eileen find the possible values of the width, x?
What is the length of the box?
What is the length of the box?
Considering the graph based on Eileen's box construction, which statements are true? (Select all that apply)
Considering the graph based on Eileen's box construction, which statements are true? (Select all that apply)
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Study Notes
Polynomial Factoring and Equations
- (2x^3y - 2y) factors completely as (2y(x - 1)(x^2 + x + 1)).
- The expression (n^3 + \frac{27}{125}) factors as ((n + \frac{3}{5})(n^2 - \frac{3}{5}n + \frac{9}{25})).
- The polynomial (121x^4 - 9y^2) can be factored as ((11x^2 - 3y)(11x^2 + 3y)).
- The polynomial equation (64x^3 + 1 = 0) has solutions (x = -\frac{1}{4}, x = 1 + \frac{i\sqrt{3}}{8}, x = 1 - \frac{i\sqrt{3}}{8}).
- Zeros of the polynomial (f(a) = a^4 - 81) include (a = -3, a = 3, a = -3i, a = 3i).
Graphical Analysis
- The domain and range of a given function are all real numbers.
- The x-intercepts are ((-3, 0)) and ( (1, 0)); the y-intercept is ((0, 3)).
- The function is positive in the intervals ((-3, 1)) and ((1, \infty)), and negative over ((-\infty, -3)).
- In another function, it decreases over ((-1.7, 1)) and increases over ((-\infty, -1.7)) and ((1, \infty)).
- The maximum value is 9.5, and the minimum is 0.
- Behavior as (x) approaches infinity: (f(x)) approaches negative infinity, while as (x) approaches negative infinity, (f(x)) approaches infinity.
Tables and Cubic Functions
- Points of a cubic function show x-intercepts at ((-3, 0), (2, 0), (3, 0)), and a y-intercept at ((0, -18)).
- As (x) approaches negative infinity, (f(x)) approaches infinity; as (x) approaches infinity, (f(x)) approaches negative infinity.
- A relative minimum occurs over the interval ( (0, -2) ) and a maximum over ((2, 3)).
Function Properties
- The function (f(x) = (x - 2)^2(x + 1)) has a y-intercept of ((0, 4)) and x-intercepts at ((-1, 0)) and ((2, 0)).
- The function approaches negative infinity as (x) approaches negative infinity and positive infinity as (x) approaches infinity.
- A relative maximum occurs between x-values (-1) and (2), with a relative minimum at ((2, 0)).
- Positive intervals include ((-1, 2)) and ((2, \infty)), while negative intervals include ((-\infty, -1)).
Box Volume and Inequalities
- For a box with volume (180 \text{ in}^3), (x) indicates length, forming the equation (x(x - 5)(x + 2) = 180).
- A cubic inequality for possible values of the width (x) is (x(x + 3.5)(x - 1.75) > 0).
- The calculated length of Eren's box is (7.52) inches.
- Eileen's box, with dimensions defined by width (x), incorporates conditions regarding its volume and overall dimensions, with the domain starting from values just greater than (1.75).
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