Polynomial Factoring and Graph Analysis

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Questions and Answers

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 $n^3 + rac{27}{125}$?

  • (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 $121x^4 - 9y^2$?

(11x^2 - 3y)(11x^2 + 3y)

What are the solutions to the polynomial equation $64x^3 + 1 = 0$?

<p>x = -1/4, x = 1 + i\sqrt{3}/8, x = 1 - i\sqrt{3}/8</p> Signup and view all the answers

What are all of the zeros of the polynomial function $f(a) = a^4 - 81$?

<p>a = -3, a = 3, a = -3i, and a = 3i</p> Signup and view all the answers

Which statement correctly fills in the blank for statement 2 to complete the proof?

<p>a(a^2 - a + 1) + 1(a^2 - a + 1)</p> Signup and view all the answers

Which polynomial expression is equal to $(2 - x)(2 + x)(4 + x^2)$?

<p>16 - x^4</p> Signup and view all the answers

Match each numbered statement with the correct reason.

<p>4 = Distributive property 5 = Combine like terms 6 = Substitute 7 = Commutative property of addition 8 = Rewrite by using the perfect square trinomial pattern 9 = Power of a product rule</p> Signup and view all the answers

If 36, 77, and 85 are the sides of a right triangle, what are the values of x and y?

<p>x = 9 and y = 2</p> Signup and view all the answers

Which statements are true about the function represented by the graph of f(x)? (Select all that apply)

<p>The domain of the function is all real numbers. (A), The range of the function is all real numbers. (B), The x-intercepts are (-3, 0) and (1, 0), and the y-intercept is (0, 3). (C), The function is positive over (-3, 1) and (1, ∞), and negative over (-∞, -3). (D)</p> Signup and view all the answers

Which statements are true about the function represented by the graph of f(x)? (Select all that apply)

<p>The function is increasing over (-∞, -1.7) and (1, ∞). (A), The relative maximum is 9.5, and the relative minimum is 0. (B), The function is decreasing over (-1.7, 1). (C), As x approaches negative infinity, f(x) approaches negative infinity. (D)</p> Signup and view all the answers

Which statements can be true about the function represented in the table? (Select all that apply)

<p>The function has a relative minimum over the interval (0, -2) and a relative maximum over the interval (2, 3). (A), The function has x-intercepts of (-3, 0), (2, 0), and (3, 0), and a y-intercept of (0, -18). (B), As x approaches -∞, f(x) approaches ∞, and as x approaches ∞, f(x) approaches -∞. (D)</p> Signup and view all the answers

Which graph represents the same function as given in the table?

<p><a href="https://assets.learnosity.com/organisations/625/asset/media/1236705">https://assets.learnosity.com/organisations/625/asset/media/1236705</a></p> Signup and view all the answers

Which statements are true about the function $f(x) = (x - 2)^2(x + 1)$? (Select all that apply)

<p>The domain is all real numbers, and the range is all real numbers. (B), The y-intercept is (0, 4). (C), The x-intercepts are (-1, 0) and (2, 0). (D)</p> Signup and view all the answers

Which statements are true about the function $f(x) = (x - 2)^2(x + 1)$? (Select all that apply)

<p>The function is positive over the intervals (-1, 2) and (2, ∞), and the function is negative over the interval (-∞, -1). (A), The function has a relative maximum between the x-values -1 and 2, and it has a relative minimum at (2, 0). (B), As x approaches -∞, f(x) approaches -∞, and as x approaches ∞, f(x) approaches ∞. (D)</p> Signup and view all the answers

Which graph represents the function that has the rule $f(x) = (x - 2)^2(x + 1)$?

<p><a href="https://assets.learnosity.com/organisations/625/asset/media/1236949">https://assets.learnosity.com/organisations/625/asset/media/1236949</a></p> Signup and view all the answers

What cubic equation would help Eren find the length of the box?

<p>x(x - 5)(x + 2) = 180</p> Signup and view all the answers

Which cubic inequality can help Eileen find the possible values of the width, x?

<p>x(x + 3.5)(x - 1.75) &gt; 0</p> Signup and view all the answers

What is the length of the box?

<p>7.52 inches</p> Signup and view all the answers

Considering the graph based on Eileen's box construction, which statements are true? (Select all that apply)

<p>The domain for the situation involves all x-values greater than 1.75. (B), To have a volume of approximately 138.13 cubic inches, the width of the box should be 5 inches. (C), The range for the situation involves all y-values greater than 0. (D)</p> Signup and view all the answers

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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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