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)?
What is the remainder when x^3 - 8x^2 + 20x - 16 is divided by x + 2?
-96
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)
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Which polynomial identities are correct? (Select all that apply)
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What is the quotient of 12x^2 - 7x + 8 divided by 3x + 2?
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What is the solution of 1/2√w + 1 = 4?
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What is the solution of √(2x - 3) + 4 = 7?
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Which expressions are equivalent to (a^2 - 16)(a + 4)? (Select the three equivalent expressions)
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Which expression shows one way to rewrite x^4 - 64?
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What is an equivalent function to y = x^2 + 8x + 10?
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What is the maximum value of the function represented by y = -4(x - 1)^2 + 3? (Fill in the blank)
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Which graph shows y = f(x + 2)?
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Which graph represents a function with a maximum value of 4?
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What represents the domain for the number of seconds a ball dropped from a height of 1,936 feet is in the air?
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Which graph represents the equation y = 2√x?
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What are the zeros of h(x)? (Select all that apply)
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As x increases without bound, h(x) ___ without bound. As x decreases without bound, h(x) ___ without bound.
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What is 3 + √-3 in the form a + bi?
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What is √-6 in the form a + bi?
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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?
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What are the solutions to the equation 0 = x^2 + 4x + 8?
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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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Description
Prepare for your Algebra 2A final exam with these flashcards. This quiz covers key concepts such as functions, remainders, and graphing equations that are essential for mastering algebra. Test your knowledge on factors, zeros, and more!
