Other Types of Equations

Polynomial Equations

Polynomial equations are equations involving polynomials, which are expressions consisting of variables raised to whole number powers and their coefficients. The general form of a polynomial equation is:

• General Form:

• Degree: The degree of a polynomial equation is the highest power of the variable in the equation.

Example 1: Solve by factoring:

• Step 1: Factor the polynomial.

• Step 2: Set each factor equal to zero and solve for .

• Step 3: Check all solutions in the original equation.

Additional info: Factoring is a common method for solving lower-degree polynomial equations, especially quadratics and cubics.

Radical Equations

Radical equations contain variables inside a radical (root) expression. To solve these equations, isolate the radical and eliminate it by raising both sides to the appropriate power.

• Solving Steps:

  1. If necessary, arrange terms so that one radical is isolated on one side of the equation.

  2. Raise both sides of the equation to the nth power to eliminate the isolated nth root.

  3. Solve the resulting equation. If this equation still contains radicals, repeat steps 1 and 2.

  4. Check all proposed solutions in the original equation to avoid extraneous solutions.

Example 2: Solve:

• Isolate the radical:

• Raise both sides to the 3rd power:

• Solve the resulting equation for .

• Check solutions in the original equation.

Equations with Rational Exponents

Equations with rational exponents involve exponents that are fractions. These can be rewritten as radical expressions and solved similarly to radical equations.

• Example 3: Solve:

• Rewrite using radicals:

• Isolate one term and solve for .

• Check solutions in the original equation.

Additional info: Rational exponents can be converted to radical form: .

Equations That Are Quadratic in Form

Some equations are not quadratic but can be rewritten in quadratic form by substitution. This method is useful for equations where the variable appears in powers that are multiples of each other.

• Example 4: Solve:

• Identify substitution: Let or as appropriate.

• Rewrite the equation in terms of .

• Solve the quadratic equation for .

• Back-substitute to solve for .

Additional info: Quadratic form is often used for equations like (let ).

Absolute Value Equations

Absolute value equations involve expressions within absolute value bars. The solution is based on the definition of absolute value.

• If is a positive real number and represents an algebraic expression, then is equivalent to or .

• Example 5: Solve:

• Set up two equations: and

• Solve each equation for .

Application: Mathematical Models

Equations can be used to model real-world situations. For example, a formula may relate weekly television viewing time to annual income.

• Example 6: The formula models weekly television viewing time (in hours) by annual income (in thousands of dollars).

• To find the annual income corresponding to 33.1 hours per week watching TV, set and solve for :

Solve for :

Additional info: Mathematical models are used to describe relationships between variables in applied contexts.