Topics in Algebra: Polynomials and Factoring

Learn to factor trinomials with leading coefficients other than one using the AC method: multiply a and c, find factors that sum to b, and factor by grouping.

Add polynomials by dropping parentheses and combining like terms with the same variable and exponent, using binomials and trinomials to produce simplified results.

Subtract polynomials by distributing a negative sign across the second polynomial, copy the first polynomial, then combine like terms to simplify, including binomial and trinomial examples.

Multiply monomials by applying the product rule: multiply coefficients and add exponents for like bases, as shown with x, y, and t in varied examples.

Apply the distributive property to multiply monomials by polynomials, using product rules for exponents to combine terms and signs, with practice problems on binomials and trinomials.

Master the foil method for multiplying binomials, using first outer inner last to form a polynomial; practice with examples to strengthen factoring and distributive property reasoning.

Use a two-distribution approach to multiply a binomial by a trinomial, an alternative to foil: multiply each binomial term by the entire second polynomial, then combine like terms.

Learn how conjugate pairs produce a difference of squares via foil, and how squaring a binomial yields a perfect square trinomial. Use foil consistently and watch middle terms.

Explore division of monomials using the quotient rule for exponents, subtract exponents for like bases, and simplify coefficients and variables, including relocating negative exponents to the denominator.

Divide polynomials by a monomial by splitting the numerator into separate fractions and simplifying each term. Use the quotient rule and cancel terms, then rewrite negative exponents as fractions.

Learn long division of polynomials to simplify rational expressions by matching first terms, using multiplication and subtraction, guided by the distributive property to obtain quotients and remainders.

Learn synthetic division for polynomials when dividing by a binomial of the form x minus c (not a catch-all), highlighting setup, coefficients, placeholding zeros, remainder handling, and reducing degree.

Explore prime factorizations by writing numbers as products, identifying prime numbers and factors, and using factor trees to reveal unique prime factorizations with examples like 24 and 18.

Identify the greatest common factor by comparing prime factorizations, as shown with 12 and 18, yielding six; extend to variables with smallest exponents like x^2 and y^2.

Identify the greatest common factor and remove it from all terms to factor polynomials, then place the factor outside the parentheses and divide each term by it, verifying by multiplication.

Factor by removing a binomial GCF, where the set of parentheses (x+y) is the GCF; divide by x+y to cancel and reveal the remaining term, with two-term expressions and foil considerations.

Master the factor by grouping method for four-term polynomials, remove any initial common factor, identify the left-hand GCF, and form a pair of binomials to complete the factoring.

Learn to factor trinomials with leading coefficient 1 using foil, including extracting a gcf, splitting x^2 into x and x, and choosing factors that sum to the middle term.

Master factoring trinomials with the AC method, handling leading coefficients other than one. Multiply a and c, factor, replace the middle term, then factor by grouping to obtain linear factors.

Identify and factor binomials as a difference of squares by recognizing conjugate pairs, using x^2 - a^2 patterns, and factoring with or without a GCF.

Remove the greatest common factor and apply patterns like difference of squares, reverse foil for trinomials, or four-term grouping or AC method; exhaust options to prove a polynomial is prime.

Identify and factor perfect square trinomials using the pattern (x+y)^2 or (x−y)^2, and weigh pattern recognition against reverse foil or the AC method when the leading coefficient is 1.

Apply a three-step factoring flowchart: remove a GCF, identify by term count, and use the appropriate method (difference of squares, AC or reverse foil, or grouping) with a final check.

Apply the zero product rule to solve quadratic equations by factoring into two factors and setting each to zero, with examples like x+4 or x-2.