Page 3 College-Level Mathematics Study Guide for the ACCUPLACER® test

Expanding Polynomials

For polynomials of degree 2 or 3, $(x + 3)^2$ or $(x - 4)^3$, it is often easiest to rewrite the expression as, $(x + 3)(x + 3)$ or $(x - 4)(x - 4)(x - 4)$, and evaluate the product of the first 2 binomials using the FOIL method (First, Outside, Inside, Last).

FOIL is an acronym to help remember how to distribute one binomial through another.

In the case of the 3rd degree polynomial, apply the FOIL method to the first 2 binomials to arrive at a trinomial and then apply the distributive property to evaluate the product of the trinomial and the last binomial:

Combine like terms to simplify the expansion:

Expanding polynomials can be thought of as the combination of the product of every term in every polynomial:

The Binomial Theorem can be useful when expanding binomials raised to larger exponents. It is the following:

For example: $(x + 2)^5 = {5\choose 0} x^{5-0}y^0 + {5\choose 1} x^{5-1}y^1 + {5\choose 2} x^{5-2}y^2 + {5\choose 3} x^{5-3}y^3 + {5\choose 4} x^{5-4}y^4 + {5\choose 5} x^{5-5}y^5$

Roots and Exponents (and their manipulation)

Expressions involving roots and exponents can be simplified and manipulated according to a set of rules.

The square root of a number is a value that when multiplied with itself returns the original number. For example, the square roots of $4$ are $2$ and $-2$ because $2 \cdot 2 = 4$ and $-2 \cdot -2 = 4$.

A number can also have a cube root. The cube root of $8$ is $2$ because $2 \cdot 2 \cdot 2 = 8$.

A number, x, multiplied with itself n times can be written as $x^n$. In this expression, x is called the base, and n is called the exponent. Only terms that are of the same exponent can be combined with addition and subtraction. This rule extends to fractional exponents, and fractional exponents are another way to write roots, for example:

Roots can be simplified by rewriting the expression inside the root as the product of factors. The expression will simplify further depending on the factors and the root. For example, $\sqrt{8}$ can be rewritten as $\sqrt{4} \cdot \sqrt{2}$. The square root of $4$ can be evaluated, so the expression becomes $2 \cdot \sqrt{2}$.

When provided with an algebraic expression involving roots and exponents, simplify roots before combining like terms. For example,

When combining exponential expressions of the same base, use the following rules:

Product of exponents: $x^a \cdot x^b = x^{a + b}$

Division of exponents: $\frac{x^a}{x^b} = x^{a - b}$

Exponent raised to an exponent: $(x^a)^b = x^{a \cdot b}$

Zero power: $x^0 = 1$ so long as $x \neq 0$

Fractional exponent: $x^{\frac{a}{b}} = \sqrt[b]{x^a}$