Algebra 2 Cumulative Exam Review

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Questions and Answers

Use the Binomial theorem to expand and where possible simplify the expression (1+i)^6, where i=√-1.

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?

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?

f(x)^-1=-1/5x-4/5 Signup and view all the answers

For the inverse variation equation , what is the value of p=8/V when V = 1/4?

D. 32 Signup and view all the answers

If (x - 5) is a factor of f(x), which of the following must be true?

B. A root of f(x) is x = 5. Signup and view all the answers

What is the remainder when (3x^3 - 2x^2 + 4x - 3) is divided by (x^2 + 3x + 3)?

D. 28x + 30 Signup and view all the answers

Which expression is equivalent to (X^-6/X^2)^3?

D. 1/X^24 Signup and view all the answers

What is the following product? Assume y≥0. 3√10(y^2√4 + √8y)

A. 6y^2√10 + 12√5y Signup and view all the answers

What is the product? 3k/k + 1 * k^2 - 1/3k^3?

B. k - 1/k^2 Signup and view all the answers

Riley makes a mistake in step 2 while doing her homework. What was her mistake?

B. She used the wrong common denominator. Signup and view all the answers

If 5 + 6i is a root of the polynomial function f(x), which of the following must also be a root of f(x)?

B. 5 - 6i Signup and view all the answers

If f(-5) = 0, what are all the factors of the function F(x)=x^3-19x+30? Use the Remainder Theorem.

A. (x - 2)(x + 5)(x - 3) Signup and view all the answers

For the inverse variation equation xy = k, what is the constant of variation, k, when x = -2 and y = 5?

A. -10 Signup and view all the answers

Which statement is true about the discontinuities of the function f(x)? f(x)= (x^2-4)/(x^3-x^2-2x)

C. There are asymptotes at x = 0 and x = -1 and a hole at (2, 2/3). Signup and view all the answers

If f(x) = 3x^2 and g(x) = 4x^3 + 1, what is the degree of (f circle g)(x)?

D. 6 Signup and view all the answers

What is the greatest possible integer value of x for which √X-5 is an imaginary number?

B. 4 Signup and view all the answers

Which statement verifies that f(x) and g(x) are inverses of each other?

D. f(g(x)) = x and g(f(x)) = x Signup and view all the answers

Which is the simplified form of (2ab/a^-5b^2)^-3?

A. b^3/8a^18 Signup and view all the answers

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?

A. 1.45 million Signup and view all the answers

What is the simplified form of the following expression? 2√27 + √12 - 3√3 - 2√12?

A. √3 Signup and view all the answers

Which second degree polynomial function has a leading coefficient of 2 and roots -3 and 5?

f(x) = 2x^2 - 4x - 30 Signup and view all the answers

Which statement about the polynomial function g(x) is true?

C. If the leading coefficient of g(x) is 1, all rational roots of g(x) = 0 must be integers. Signup and view all the answers

Which solution to the equation 3/a + 2 + 2/a = 4a - 4/a^2 - 4 is extraneous?

A. -2 Signup and view all the answers

What is the following quotient? 6 - 3(√6)/ (√9)?

A. 2(√3 - √18) Signup and view all the answers

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.