Adding and subtracting radicals works like combining like terms. You can only combine like radicals, which have the same number under the radical sign. Add or subtract the numbers in front, and keep the radical the same. For example, 3√5 + 4√5 = 7√5.
Simplifying square roots means rewriting a square root so no perfect square factor is left under the radical sign. Find the largest perfect square that divides the number, split it out, and take its square root. For example, √72 = √(36 · 2) = 6√2. This is called simplest radical form.
Factoring completely means writing a polynomial as a product of factors that can’t be factored any more. Take out the GCF first. Then use the number of terms to pick a method, and keep going until every factor is prime. For example, 2x² − 18 = 2(x + 3)(x − 3).
A perfect square trinomial is a trinomial that equals a binomial squared. It fits a² + 2ab + b² = (a + b)² or a² − 2ab + b² = (a − b)². The first and last terms are perfect squares, and the middle term is 2 times their roots. For example, x² + 10x + 25 = (x + 5)².
Factoring the difference of squares means rewriting a square minus a square as a sum times a difference. The rule is a² − b² = (a + b)(a − b). Find what is squared in each term, then write one plus and one minus. For example, x² − 25 = (x + 5)(x − 5). It is a special product run backward.
Factoring by grouping is a way to factor a polynomial with four terms. Group the terms in pairs, factor the GCF out of each pair, and then factor out the binomial they share. For example, x³ + 3x² + 2x + 6 = x²(x + 3) + 2(x + 3) = (x + 3)(x² + 2). It also finishes the AC method for trinomials.
Factoring trinomials when a is not 1 means factoring ax² + bx + c when the leading coefficient a is not 1. Use the AC method: multiply a and c, find two numbers that multiply to ac and add to b, then split the middle term and use a box. For example, 3x² + 10x + 8 = (3x + 4)(x + 2).
Factoring trinomials of the form x² + bx + c means writing them as a product of two binomials, (x + p)(x + q). Find two numbers that multiply to c and add to b. For example, x² + 7x + 12 = (x + 3)(x + 4), because 3 · 4 = 12 and 3 + 4 = 7. This works when the number in front of x² is 1.
Factoring out the GCF means finding the greatest common factor of every term in a polynomial and writing it in front of parentheses. It is the distributive property run backward. For example, 6x² + 9x = 3x(2x + 3), because 3x divides both terms.
Special products of binomials are shortcuts for three products that show up all the time: (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², and (a + b)(a − b) = a² − b². Learn the patterns and you can skip FOIL.
