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.
A geometric sequence is a list of numbers where you multiply by the same number each time to get the next term. That number is the common ratio, r. For example, 2, 6, 18, 54, … multiplies by 3. The formula aₙ = a₁ · rⁿ⁻¹ finds any term.
Compound interest is interest earned on both the money you put in and the interest it has already earned. The formula is A = P(1 + r/n)ⁿᵗ, where P is the starting amount, r is the yearly rate as a decimal, n is how many times a year interest is added, and t is the number of years.
Exponential growth and decay describe amounts that change by the same percent each time period. For growth, use y = a(1 + r)ᵗ. For decay, use y = a(1 − r)ᵗ. Here a is the starting amount, r is the rate as a decimal, and t is the number of time periods.
An exponential function has the variable in the exponent. Its form is f(x) = a · bˣ, where a is the starting value and b is the number you multiply by each time x goes up by 1. For example, f(x) = 3 · 2ˣ doubles at every step.
Zero and negative exponents follow two rules. Any nonzero number to the zero power is 1, so 7⁰ = 1. A negative exponent means “one over” the positive power, so x⁻ⁿ = 1/xⁿ. For example, 2⁻³ = 1/8. A negative exponent never makes the number negative.
