Precalculus Test Review: Equations, Inequalities, and Polynomial Functions

Equations and Inequalities

Solving Quadratic Inequalities

Quadratic inequalities involve expressions of the form ax2 + bx + c > 0 or similar. Solving these requires finding the roots and testing intervals.

• Key Point 1: Set the quadratic equal to zero and solve for the roots.

• Key Point 2: Use a number line and sign chart to determine where the inequality holds.

• Example: Solve $x^2 - 3x + 2 > 0$ algebraically and graphically.

  • Factor: $x^2 - 3x + 2 = (x-1)(x-2)$

  • Roots: $x = 1$, $x = 2$

  • Test intervals: $(-, 1)$, $(1, 2)$, $(2, )$

  • Sign chart: Positive outside roots, negative between roots.

Solving Rational Inequalities

Rational inequalities involve expressions where a polynomial is divided by another polynomial.

• Key Point 1: Set the numerator and denominator equal to zero to find critical points.

• Key Point 2: Use a sign chart to test intervals between critical points.

• Example: Solve $\frac{x^2}{x-2} \geq 0$.

  • Critical points: $x = 0$, $x = 2$

  • Test intervals: $(-, 0)$, $(0, 2)$, $(2, )$

  • Check sign of numerator and denominator in each interval.

Solving Linear Equations

Linear equations are solved by isolating the variable using algebraic operations.

• Key Point: Use inverse operations to solve for the variable.

• Example: Solve $ax = b$ for $x$.

Polynomial Functions

Roots and Solutions of Polynomial Equations

Finding the roots of a polynomial involves factoring, using the quadratic formula, or other algebraic methods.

• Key Point 1: The Fundamental Theorem of Algebra states that a polynomial of degree $n$ has $n$ roots (real or complex).

• Key Point 2: Roots can be real or complex; use factoring, synthetic division, or the quadratic formula as appropriate.

• Example: Find all solutions to $x^3 - 2x^2 + x - 2 = 0$.

  • Try rational root theorem, factor, or synthetic division.

Constructing Polynomials from Roots

Given roots, you can construct a polynomial by multiplying factors of the form $(x - r)$ for each root $r$.

• Key Point: For roots $a$, $b$, $c$, the polynomial is $(x - a)(x - b)(x - c)$.

• Example: Find a third degree polynomial with roots $x = 0$, $x = 2$, $x = -1$:

  • Polynomial: $(x)(x-2)(x+1)$

Multiplicity of Roots

The multiplicity of a root refers to how many times a particular root appears in the factorization of a polynomial.

• Key Point: If $(x - r)^k$ is a factor, $r$ is a root of multiplicity $k$.

• Example: $x^3 - 3x^2 + 3x - 1 = (x-1)^3$; root $x=1$ has multiplicity 3.

Real and Complex Roots Table

Polynomials can have both real and complex roots. The following table summarizes the classification:

Root Type	Description
Real	Roots that are real numbers; can be found by factoring or using the quadratic formula.
Complex	Roots that involve imaginary numbers; occur when the discriminant is negative.

Synthetic Division and the Remainder Theorem

Synthetic division is a shortcut method for dividing a polynomial by a linear factor. The Remainder Theorem states that the remainder of $f(x)$ divided by $(x - a)$ is $f(a)$.

• Key Point: Use synthetic division to quickly divide polynomials and find remainders.

• Example: Given $f(x) = 2x^3 - 5x^2 + 10x$, find $f(1)$ using synthetic division.

Fundamental Concepts of Algebra

Simplifying Radical Expressions

Radical expressions can be simplified by factoring under the radical and using properties of exponents.

• Key Point: $\sqrt{a^2} = |a|$ and $\sqrt{ab} = \sqrt{a} \sqrt{b}$

• Example: Simplify $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$

Operations with Exponents

Exponents follow specific rules for multiplication, division, and powers.

• Key Point: $a^m \cdot a^n = a^{m+n}$, $\frac{a^m}{a^n} = a^{m-n}$, $(a^m)^n = a^{mn}$

• Example: Simplify $a^2 \cdot a^3 = a^{2+3} = a^5$

Factoring Polynomials

Factoring is the process of writing a polynomial as a product of its factors.

• Key Point: Common methods include factoring by grouping, using the quadratic formula, and recognizing special products.

• Example: Factor $x^2 - 9 = (x-3)(x+3)$

Divisibility and Factors

Determining if one polynomial is a factor of another involves division and checking for a remainder of zero.

• Key Point: If $f(x)$ divided by $g(x)$ leaves no remainder, then $g(x)$ is a factor of $f(x)$.

• Example: Is $x-2$ a factor of $x^2 - 2x + 4$? Substitute $x=2$; if $f(2)=0$, then yes.

Additional info:

• Some questions reference sign charts and number lines, which are standard tools for analyzing inequalities.

• Questions cover synthetic division and the Remainder Theorem, which are essential for polynomial division and root analysis.

• Problems include constructing polynomials from given roots, a key skill in Precalculus.