Algebra 2
Polynomial Operations and Complex Numbers flashcards
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- In Polynomial Arithmetic, what does like-term addition mean?Polynomial Arithmetic
- Combining coefficients on matching powers.
- Can an x term combine with an x² term?Polynomial Arithmetic
- No, their variable parts differ.
- In Polynomial Arithmetic, what does distribution and products mean?Polynomial Arithmetic
- A polynomial product formed by repeated distribution.
- Why combine middle terms after expansion?Polynomial Arithmetic
- They represent the same variable power.
- In Polynomial Arithmetic, what does special products mean?Polynomial Arithmetic
- Recognizable products with specific factorizations.
- Why does (u+v)² include 2uv?Polynomial Arithmetic
- There are two uv cross-products.
- In Polynomial Division, what does long division mean?Polynomial Division
- An algorithm giving quotient and remainder.
- What stops the repeated steps?Polynomial Division
- A remainder whose degree is below the divisor degree.
- In Polynomial Division, what does synthetic division mean?Polynomial Division
- A streamlined division algorithm for x−r.
- Why insert zero coefficients?Polynomial Division
- They keep powers correctly aligned.
- In Polynomial Division, what does quotient-remainder identity mean?Polynomial Division
- The defining check for polynomial division.
- How do you verify a quotient?Polynomial Division
- Multiply divisor by quotient and add the remainder.
- In Factor and Remainder Theorems, what does remainder theorem mean?Factor and Remainder Theorems
- A link between evaluation and division remainder.
- What input is used for divisor x−r?Factor and Remainder Theorems
- Evaluate at x=r.
- In Factor and Remainder Theorems, what does factor theorem mean?Factor and Remainder Theorems
- The zero-remainder criterion for a factor.
- How can a false candidate be rejected quickly?Factor and Remainder Theorems
- Evaluate p(r) and see whether it is nonzero.
- In Factor and Remainder Theorems, what does factoring after division mean?Factor and Remainder Theorems
- Reducing polynomial degree by a known factor.
- Why compare root multiplicities with degree?Factor and Remainder Theorems
- The total multiplicity of factors should match the degree over complex numbers.
- In Complex Numbers, what does imaginary unit mean?Complex Numbers
- A number with real and imaginary components.
- Is i itself a real number?Complex Numbers
- No, no real square equals −1.
- In Complex Numbers, what does complex arithmetic mean?Complex Numbers
- Operations on numbers a+bi.
- Why replace i² in a product?Complex Numbers
- By definition i²=−1.
- In Complex Numbers, what does conjugates and powers mean?Complex Numbers
- A conjugate pair and repeating imaginary-unit powers.
- Why does i⁴ equal 1?Complex Numbers
- Because (i²)²=(−1)²=1.
- In Solving Polynomial Equations, what does real roots mean?Solving Polynomial Equations
- Real values making a polynomial equal zero.
- What does a graph x-intercept represent?Solving Polynomial Equations
- A real root of the polynomial equation.
- In Solving Polynomial Equations, what does nonreal roots mean?Solving Polynomial Equations
- Roots outside the real number line.
- Does no real intercept mean no complex roots?Solving Polynomial Equations
- No, nonconstant polynomials have complex roots counting multiplicity.
- In Solving Polynomial Equations, what does multiplicity and checks mean?Solving Polynomial Equations
- The count of a root within a factorization.
- Why check the original equation?Solving Polynomial Equations
- Division or algebraic manipulation can introduce errors.