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.