Equivalent Expressions and Algebraic Structure flashcards

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Subtract compound polynomialsCombining Terms and Polynomial Operations: Change every sign in the polynomial being subtracted.
When should I use collect polynomial terms?Combining Terms and Polynomial Operations: Combine coefficients of like powers.
What method supports subtract compound polynomials?Combining Terms and Polynomial Operations: P − (Q + R) = P − Q − R
Multiply polynomialsCombining Terms and Polynomial Operations: Every term in the first group multiplies every term in the second.
Collect polynomial termsCombining Terms and Polynomial Operations: Match the powers before adding the front numbers.
What must I check in combining terms and polynomial operations?Combining Terms and Polynomial Operations: Combine terms only when their variable factors and exponents match
What method supports multiply polynomials?Combining Terms and Polynomial Operations: (x + a)(x + b) = x² + (a + b)x + ab
What method supports collect polynomial terms?Combining Terms and Polynomial Operations: ax² + bx² = (a + b)x²
When should I use subtract compound polynomials?Combining Terms and Polynomial Operations: Preserve signs when subtracting one whole expression from another.
When should I use multiply polynomials?Combining Terms and Polynomial Operations: Use distribution to find a quadratic coefficient.
Perfect-square structureFactoring Common Factors and Special Products: The middle coefficient is twice the product, not the square of the constant.
When should I use common-factor extraction?Factoring Common Factors and Special Products: Factor out the greatest common numerical and variable factor.
What method supports perfect-square structure?Factoring Common Factors and Special Products: (ax + b)² = a²x² + 2abx + b²
Difference of squaresFactoring Common Factors and Special Products: Two squares separated by subtraction produce conjugate factors.
Common-factor extractionFactoring Common Factors and Special Products: Take out what every term shares.
What must I check in factoring common factors and special products?Factoring Common Factors and Special Products: Factorization must reproduce every original term; the greatest common factor includes shared variable powers
What method supports difference of squares?Factoring Common Factors and Special Products: A² − B² = (A − B)(A + B)
What method supports common-factor extraction?Factoring Common Factors and Special Products: ab + ac = a(b + c)
When should I use perfect-square structure?Factoring Common Factors and Special Products: Recover a missing coefficient in a perfect-square trinomial.
When should I use difference of squares?Factoring Common Factors and Special Products: Recognize and factor a difference of two squares.
Factoring after extractionFactoring Quadratic Expressions: Take out the shared factor before searching for binomials.
When should I use monic quadratic factoring?Factoring Quadratic Expressions: Find two integers with a specified sum and product.
What method supports factoring after extraction?Factoring Quadratic Expressions: k(x² − (r + s)x + rs) = k(x − r)(x − s)
Factoring with a leading coefficientFactoring Quadratic Expressions: Check the two cross products, then add them.
Monic quadratic factoringFactoring Quadratic Expressions: Find two numbers that both add and multiply correctly.
What must I check in factoring quadratic expressions?Factoring Quadratic Expressions: A factored identity must match both the linear coefficient and constant term
What method supports factoring with a leading coefficient?Factoring Quadratic Expressions: (ax + r)(bx + s) = abx² + (as + br)x + rs
What method supports monic quadratic factoring?Factoring Quadratic Expressions: x² + (r + s)x + rs = (x + r)(x + s)
When should I use factoring after extraction?Factoring Quadratic Expressions: Remove a common factor before interpreting a quadratic factor.
When should I use factoring with a leading coefficient?Factoring Quadratic Expressions: Use products of binomials when the leading coefficient is not one.
Equivalent exponential powersExponent Rules and Rational Exponents: Rewrite different-looking bases using the same foundation.
When should I use products and quotients of powers?Exponent Rules and Rational Exponents: Combine equal-base powers by exponent laws.
What method supports equivalent exponential powers?Exponent Rules and Rational Exponents: (a^p)^q = a^(pq)
Rational powersExponent Rules and Rational Exponents: The denominator of the exponent tells which root to take.
Products and quotients of powersExponent Rules and Rational Exponents: Add powers when multiplying; subtract them when dividing.
What must I check in exponent rules and rational exponents?Exponent Rules and Rational Exponents: For real rational powers, even-denominator roots require nonnegative radicands; a negative power excludes zero
What method supports rational powers?Exponent Rules and Rational Exponents: a^(m/n) = (ⁿ√a)^m
What method supports products and quotients of powers?Exponent Rules and Rational Exponents: a^m a^n = a^(m+n); a^m/a^n = a^(m−n), a ≠ 0
When should I use equivalent exponential powers?Exponent Rules and Rational Exponents: Use a common base to compare structured exponents.
When should I use rational powers?Exponent Rules and Rational Exponents: Interpret a fractional exponent as a root followed by a power.
Radicals with variables and signsSimplifying Radical Expressions: An even root reports a nonnegative magnitude.
When should I use extract perfect squares?Simplifying Radical Expressions: Rewrite a square root using a perfect-square factor.
What method supports radicals with variables and signs?Simplifying Radical Expressions: For x < 0, √(x²) = −x
Combine like radicalsSimplifying Radical Expressions: Make the inside match before combining the front numbers.
Extract perfect squaresSimplifying Radical Expressions: Pull out the square factors and leave the rest inside.
What must I check in simplifying radical expressions?Simplifying Radical Expressions: For real square roots the radicand must be nonnegative; √(x²) = |x|
What method supports combine like radicals?Simplifying Radical Expressions: a√r + b√r = (a + b)√r
What method supports extract perfect squares?Simplifying Radical Expressions: √(a²b) = |a|√b, b ≥ 0
When should I use radicals with variables and signs?Simplifying Radical Expressions: Use absolute value when extracting an even power.
When should I use combine like radicals?Simplifying Radical Expressions: Add radical terms only after simplifying their radicands.
Combining rational expressionsRational Expressions and Domain Restrictions: Build one fraction, then factor if possible.
When should I use canceled factors and holes?Rational Expressions and Domain Restrictions: Simplify a rational expression while retaining its original restrictions.
What method supports combining rational expressions?Rational Expressions and Domain Restrictions: a/x + b/x = (a + b)/x, x ≠ 0
Domain exclusionsRational Expressions and Domain Restrictions: Only denominator zeros make a rational expression undefined.
Canceled factors and holesRational Expressions and Domain Restrictions: Cancel factors, then keep the original forbidden inputs.
What must I check in rational expressions and domain restrictions?Rational Expressions and Domain Restrictions: Original denominators remain excluded even if a common factor cancels
What method supports domain exclusions?Rational Expressions and Domain Restrictions: (x − a)(x + b) ≠ 0 ⇒ x ≠ a and x ≠ −b
What method supports canceled factors and holes?Rational Expressions and Domain Restrictions: (x² − a²)/(x − a) = x + a for x ≠ a
When should I use combining rational expressions?Rational Expressions and Domain Restrictions: Use common denominators and preserve restrictions.
When should I use domain exclusions?Rational Expressions and Domain Restrictions: Find all zeros of an original factored denominator.