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Questions and Answers
If f(2) = 0 for the function f(x) = 2x^3 − 9x^2 − 15x + 50, which of the following must be a factor of f(x)?
If f(2) = 0 for the function f(x) = 2x^3 − 9x^2 − 15x + 50, which of the following must be a factor of f(x)?
- x + 2
- x^2 + 1
- x^3 - 3x + 4
- x - 2 (correct)
What is the remainder when x^3 - 8x^2 + 20x - 16 is divided by x + 2?
What is the remainder when x^3 - 8x^2 + 20x - 16 is divided by x + 2?
-96
Match the correct graph with each equation.
Match the correct graph with each equation.
y = x^2 = Parabola opening upwards y = -x^2 = Parabola opening downwards y = |x| = V-shaped graph
What are the zeros of the function f(x) = x^3 − 36x^2 + 35x? (Select all that apply)
What are the zeros of the function f(x) = x^3 − 36x^2 + 35x? (Select all that apply)
Which polynomial identities are correct? (Select all that apply)
Which polynomial identities are correct? (Select all that apply)
What is the quotient of 12x^2 - 7x + 8 divided by 3x + 2?
What is the quotient of 12x^2 - 7x + 8 divided by 3x + 2?
What is the solution of 1/2√w + 1 = 4?
What is the solution of 1/2√w + 1 = 4?
What is the solution of √(2x - 3) + 4 = 7?
What is the solution of √(2x - 3) + 4 = 7?
Which expressions are equivalent to (a^2 - 16)(a + 4)? (Select the three equivalent expressions)
Which expressions are equivalent to (a^2 - 16)(a + 4)? (Select the three equivalent expressions)
Which expression shows one way to rewrite x^4 - 64?
Which expression shows one way to rewrite x^4 - 64?
What is an equivalent function to y = x^2 + 8x + 10?
What is an equivalent function to y = x^2 + 8x + 10?
What is the maximum value of the function represented by y = -4(x - 1)^2 + 3? (Fill in the blank)
What is the maximum value of the function represented by y = -4(x - 1)^2 + 3? (Fill in the blank)
Which graph shows y = f(x + 2)?
Which graph shows y = f(x + 2)?
Which graph represents a function with a maximum value of 4?
Which graph represents a function with a maximum value of 4?
What represents the domain for the number of seconds a ball dropped from a height of 1,936 feet is in the air?
What represents the domain for the number of seconds a ball dropped from a height of 1,936 feet is in the air?
Which graph represents the equation y = 2√x?
Which graph represents the equation y = 2√x?
What are the zeros of h(x)? (Select all that apply)
What are the zeros of h(x)? (Select all that apply)
As x increases without bound, h(x) ___ without bound. As x decreases without bound, h(x) ___ without bound.
As x increases without bound, h(x) ___ without bound. As x decreases without bound, h(x) ___ without bound.
What is 3 + √-3 in the form a + bi?
What is 3 + √-3 in the form a + bi?
What is √-6 in the form a + bi?
What is √-6 in the form a + bi?
The expression (3 + i)(1 + 2i) can be written in the form a + bi, where a and b are integers. What are the values of a and b?
The expression (3 + i)(1 + 2i) can be written in the form a + bi, where a and b are integers. What are the values of a and b?
What are the solutions to the equation 0 = x^2 + 4x + 8?
What are the solutions to the equation 0 = x^2 + 4x + 8?
Solve the equation 2x^3 = -20. Select all solutions of the equation.
Solve the equation 2x^3 = -20. Select all solutions of the equation.
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Study Notes
Polynomial Functions and Factors
- For the function f(x) = 2x^3 − 9x^2 − 15x + 50, if f(2) = 0, then x - 2 is a factor of f(x).
- The Remainder Theorem states that the remainder of the division of a polynomial by x + c is equal to f(-c). For f(x) = x^3 - 8x^2 + 20x - 16 divided by x + 2, the remainder is -96.
Roots and Solutions
- Zeros of the function f(x) = x^3 − 36x^2 + 35x include 0, 1, and 35.
- The equation 2x^3 = -20 gives solutions x = √10i and x = -√10i, indicating complex roots.
Polynomial Identities
- Correct polynomial identities include:
- (a − b)(a + b) = a^2 − b^2
- (a^2 + b^2)^2 = a^4 + 2a^2b^2 + b^4
Quotients and Rewrites
- Dividing 12x^2 - 7x + 8 by 3x + 2 results in the quotient 4x - 5 + 18/(3x + 2).
- The expression x^4 - 64 can be rewritten as (x^2 + 8)(x^2 - 8).
Transformations and Maximum Values
- The equivalent function for y = x^2 + 8x + 10 is y = (x + 4)^2 - 6.
- The function y = -4(x - 1)^2 + 3 has a maximum value of 3 at the point (1, 3).
Graphs and Functions
- The function y = f(x + 2) requires identifying the correct graph transformation.
- A specific graph indicates a function with a maximum value of 4.
- The graph y = 2√x displays the square root function scaled by a factor of 2.
Complex Numbers
- The expression 3 + √-3 can be represented as 3 + √3i, indicating a real part and an imaginary part.
- √-6 can be expressed as 0 + √6i in the form a + bi, confirming its imaginary nature.
Ball Drop Problem
- The height of a ball dropped from 1,936 feet is modeled by h(t) = −16t^2 + 1,936, with its domain covering all real numbers (t ≥ 0).
Additional Key Details
- The values of a and b for the expression (3 + i)(1 + 2i) are a = 1 and b = 7, showing how to convert complex products into standard form.
- The equation 0 = x^2 + 4x + 8 has solutions x = −2 ± 2i, highlighting the nature of quadratic equations yielding complex solutions.
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