Podcast
Questions and Answers
Use the Binomial theorem to expand and where possible simplify the expression (1+i)^6, where i=√-1.
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.
Find the 5th term of the expansion of (x + y)9.
- 126x^5y^4 (correct)
- 81x^4y^5
- 81x^5y^4
- 126x^4y^5
Which second degree polynomial function f(x) has a lead coefficient of 3 and roots 4 and 1?
Which second degree polynomial function f(x) has a lead coefficient of 3 and roots 4 and 1?
- f(x) = 3x^2 + 15x + 12
- f(x) = 3x^2 + 12x + 4 (correct)
- f(x) = 3x^2 - 15x + 12
- f(x) = 3x^2 - 12x + 4
Which function is the inverse of f(x)=-5x-4?
Which function is the inverse of f(x)=-5x-4?
For the inverse variation equation , what is the value of p=8/V when V = 1/4?
For the inverse variation equation , what is the value of p=8/V when V = 1/4?
If (x - 5) is a factor of f(x), which of the following must be true?
If (x - 5) is a factor of f(x), which of the following must be true?
What is the remainder when (3x^3 - 2x^2 + 4x - 3) is divided by (x^2 + 3x + 3)?
What is the remainder when (3x^3 - 2x^2 + 4x - 3) is divided by (x^2 + 3x + 3)?
Which expression is equivalent to (X^-6/X^2)^3?
Which expression is equivalent to (X^-6/X^2)^3?
What is the following product? Assume y≥0. 3√10(y^2√4 + √8y)
What is the following product? Assume y≥0. 3√10(y^2√4 + √8y)
What is the product? 3k/k + 1 * k^2 - 1/3k^3?
What is the product? 3k/k + 1 * k^2 - 1/3k^3?
Riley makes a mistake in step 2 while doing her homework. What was her mistake?
Riley makes a mistake in step 2 while doing her homework. What was her mistake?
If 5 + 6i is a root of the polynomial function f(x), which of the following must also be a root of f(x)?
If 5 + 6i is a root of the polynomial function f(x), which of the following must also be a root of f(x)?
If f(-5) = 0, what are all the factors of the function F(x)=x^3-19x+30? Use the Remainder Theorem.
If f(-5) = 0, what are all the factors of the function F(x)=x^3-19x+30? Use the Remainder Theorem.
For the inverse variation equation xy = k, what is the constant of variation, k, when x = -2 and y = 5?
For the inverse variation equation xy = k, what is the constant of variation, k, when x = -2 and y = 5?
Which statement is true about the discontinuities of the function f(x)? f(x)= (x^2-4)/(x^3-x^2-2x)
Which statement is true about the discontinuities of the function f(x)? f(x)= (x^2-4)/(x^3-x^2-2x)
If f(x) = 3x^2 and g(x) = 4x^3 + 1, what is the degree of (f circle g)(x)?
If f(x) = 3x^2 and g(x) = 4x^3 + 1, what is the degree of (f circle g)(x)?
What is the greatest possible integer value of x for which √X-5 is an imaginary number?
What is the greatest possible integer value of x for which √X-5 is an imaginary number?
Which statement verifies that f(x) and g(x) are inverses of each other?
Which statement verifies that f(x) and g(x) are inverses of each other?
Which is the simplified form of (2ab/a^-5b^2)^-3?
Which is the simplified form of (2ab/a^-5b^2)^-3?
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?
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?
What is the simplified form of the following expression? 2√27 + √12 - 3√3 - 2√12?
What is the simplified form of the following expression? 2√27 + √12 - 3√3 - 2√12?
Which second degree polynomial function has a leading coefficient of 2 and roots -3 and 5?
Which second degree polynomial function has a leading coefficient of 2 and roots -3 and 5?
Which statement about the polynomial function g(x) is true?
Which statement about the polynomial function g(x) is true?
Which solution to the equation 3/a + 2 + 2/a = 4a - 4/a^2 - 4 is extraneous?
Which solution to the equation 3/a + 2 + 2/a = 4a - 4/a^2 - 4 is extraneous?
What is the following quotient? 6 - 3(√6)/ (√9)?
What is the following quotient? 6 - 3(√6)/ (√9)?
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}) ).
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
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.