Questions and Answers
Use the Binomial theorem to expand and where possible simplify the expression (1+i)^6, where i=√-1.
A
Find the 5th term of the expansion of (x + y)9.
Which second degree polynomial function f(x) has a lead coefficient of 3 and roots 4 and 1?
Which function is the inverse of f(x)=-5x-4?
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For the inverse variation equation , what is the value of p=8/V when V = 1/4?
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If (x - 5) is a factor of f(x), which of the following must be true?
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What is the remainder when (3x^3 - 2x^2 + 4x - 3) is divided by (x^2 + 3x + 3)?
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Which expression is equivalent to (X^-6/X^2)^3?
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What is the following product? Assume y≥0. 3√10(y^2√4 + √8y)
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What is the product? 3k/k + 1 * k^2 - 1/3k^3?
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Riley makes a mistake in step 2 while doing her homework. What was her mistake?
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If 5 + 6i is a root of the polynomial function f(x), which of the following must also be a root of f(x)?
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If f(-5) = 0, what are all the factors of the function F(x)=x^3-19x+30? Use the Remainder Theorem.
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For the inverse variation equation xy = k, what is the constant of variation, k, when x = -2 and y = 5?
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Which statement is true about the discontinuities of the function f(x)? f(x)= (x^2-4)/(x^3-x^2-2x)
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If f(x) = 3x^2 and g(x) = 4x^3 + 1, what is the degree of (f circle g)(x)?
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What is the greatest possible integer value of x for which √X-5 is an imaginary number?
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Which statement verifies that f(x) and g(x) are inverses of each other?
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Which is the simplified form of (2ab/a^-5b^2)^-3?
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The price that a company charged for a computer accessory is given by the equation where x is the number of accessories that are produced, in millions. It costs the company $10 to make each accessory. The company currently produces 2 million accessories and makes a profit of 100 million dollars. What other number of accessories produced yields approximately the same profit?
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What is the simplified form of the following expression? 2√27 + √12 - 3√3 - 2√12?
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Which second degree polynomial function has a leading coefficient of 2 and roots -3 and 5?
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Which statement about the polynomial function g(x) is true?
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Which solution to the equation 3/a + 2 + 2/a = 4a - 4/a^2 - 4 is extraneous?
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What is the following quotient? 6 - 3(√6)/ (√9)?
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Study Notes
Binomial Theorem and Expansion
- The Binomial Theorem provides a method for expanding expressions of the form ( (a + b)^n ).
- For ( (1 + i)^6 ), applying the theorem gives complex results involving powers of ( i ).
Polynomial Expansion
- The 5th term of the expansion of ( (x + y)^9 ) corresponds to ( 126x^5y^4 ).
Polynomial Functions
- A second-degree polynomial with a leading coefficient of 3 and roots at 4 and 1 can be expressed as ( f(x) = 3(x - 4)(x - 1) ) which simplifies to ( 3x^2 - 15x + 12 ).
Inverse Functions
- The inverse of the function ( f(x) = -5x - 4 ) is ( f^{-1}(x) = -\frac{1}{5}x - \frac{4}{5} ).
Inverse Variation
- In the equation ( p = \frac{8}{V} ), when ( V = \frac{1}{4} ), the value of ( p ) computes to 32.
Factors and Roots
- If ( (x - 5) ) is a factor of ( f(x) ), then ( x = 5 ) must be a root of ( f(x) ).
Remainder and Polynomial Division
- When dividing ( 3x^3 - 2x^2 + 4x - 3 ) by ( x^2 + 3x + 3 ), the remainder is ( 28x + 30 ).
Simplification of Expressions
- The expression ( \left(\frac{X^{-6}}{X^2}\right)^3 ) simplifies to ( \frac{1}{X^{24}} ).
- The product ( 3\sqrt{10}(y^2\sqrt{4} + \sqrt{8y}) ) simplifies to ( 6y^2\sqrt{10} + 12\sqrt{5}y ).
Fraction Simplification
- The product ( \frac{3k}{k + 1} \cdot \frac{k^2 - 1}{3k^3} ) simplifies to ( \frac{k - 1}{k^2} ).
Common Errors in Algebra
- Using the wrong common denominator can lead to mistakes in calculations.
Complex Roots
- If ( 5 + 6i ) is a root of a polynomial, then its conjugate ( 5 - 6i ) is also a root.
Remainder Theorem Application
- If ( f(-5) = 0 ) for ( F(x) = x^3 - 19x + 30 ), then the factors are ( (x - 2)(x + 5)(x - 3) ).
Constants of Variation
- For the inverse variation ( xy = k ), when ( x = -2 ) and ( y = 5 ), the constant ( k ) is -10.
Discontinuities in Functions
- The function ( f(x) = \frac{x^2 - 4}{x^3 - x^2 - 2x} ) has asymptotes at ( x = 0 ) and ( x = -1 ) with a hole at ( (2, \frac{2}{3}) ).
Function Composition Degrees
- The degree of the composition ( (f \circ g)(x) ), with ( f(x) = 3x^2 ) and ( g(x) = 4x^3 + 1 ), is 6.
Conditions for Imaginary Numbers
- The greatest integer value of ( x ) such that ( \sqrt{x - 5} ) is imaginary is 4.
Verification of Inverses
- Functions ( f(x) ) and ( g(x) ) are inverses if ( f(g(x)) = x ) and ( g(f(x)) = x ).
Simplification of Algebraic Expressions
- The expression ( \left(\frac{2ab}{a^{-5}b^2}\right)^{-3} ) simplifies to ( \frac{b^3}{8a^{18}} ).
Profit Maximization Scenario
- A company producing 2 million accessories makes a profit of 100 million dollars, and producing approximately 1.45 million accessories yields a similar profit.
Simplifying Radical Expressions
- The expression ( 2\sqrt{27} + \sqrt{12} - 3\sqrt{3} - 2\sqrt{12} ) simplifies to ( \sqrt{3} ).
Characteristics of Polynomial Functions
- For a polynomial function ( g(x) ) with a leading coefficient of 1, all rational roots must be integers.
Identification of Extraneous Solutions
- In the equation ( \frac{3}{a} + 2 + \frac{2}{a} = \frac{4a - 4}{a^2 - 4} ), the extraneous solution is ( -2 ).
Quotient of Roots
- The quotient ( \frac{6 - 3\sqrt[3]{6}}{\sqrt[3]{9}} ) simplifies to ( 2(\sqrt[3]{3} - \sqrt[3]{18}) ).
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Prepare for your Algebra 2 exam with these cumulative review flashcards. Each card covers essential concepts such as the Binomial theorem, polynomial functions, and term expansions. Test your understanding and readiness for the upcoming assessment.
