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.
Multiplying polynomials means multiplying every term of one polynomial by every term of the other, then combining like terms. You can use the distributive property or a box (grid) to keep track of the products. For example, (x + 2)(x² + 3x − 1) = x³ + 5x² + 5x − 2.
Multiplying binomials means multiplying each term of the first binomial by each term of the second, then combining like terms. The FOIL method helps you remember all four products: First, Outer, Inner, Last. For example, (x + 3)(x + 5) = x² + 8x + 15.
Adding and subtracting polynomials comes down to combining like terms. To add, drop the parentheses and combine terms with the same variable and exponent. To subtract, change the sign of every term in the second polynomial, then add.
A polynomial is an expression made of one or more terms added or subtracted. Each term is a number times variables with whole-number exponents, like 4x³ − 2x + 7. Polynomials are named by their number of terms and their degree, which is the highest exponent.
To compare linear vs. exponential functions, look at how the output changes each step. A linear function adds the same amount each time, so its graph is a straight line. An exponential function multiplies by the same number each time, so its graph curves.