Equations and Inequalities

Solving Polynomial Equations by Factoring

Polynomial equations are equations involving polynomials set equal to another expression, often zero. The degree of a polynomial equation is determined by the highest power of the variable present. Solving these equations often involves factoring and applying the zero-product principle.

• Zero-Product Principle: If a product of factors equals zero, then at least one of the factors must be zero.

• Factoring: Express the polynomial as a product of simpler polynomials.

• Solving: Set each factor equal to zero and solve for the variable.

• Example: Solve by factoring:

  • Factor:

  • Set each factor to zero: or

  • Solutions: or

Additional info: The degree of a polynomial equation determines the maximum number of solutions it can have. For example, a degree 1 equation (linear) has one solution, degree 2 (quadratic) has up to two, and degree 3 (cubic) has up to three.

Solving Radical Equations

Radical equations contain variables within a root. When solving radical equations, especially those with even roots, it is important to check all proposed solutions, as extraneous solutions may arise from the process of squaring both sides.

• Steps to Solve:

  1. Isolate the radical expression.

  2. Raise both sides to the power that eliminates the radical.

  3. Solve the resulting equation.

  4. Check all solutions in the original equation.

• Example: Solve

  • Square both sides:

  • Expand:

  • Rearrange:

  • Factor:

  • Solutions: or

  • Check in original equation to confirm validity.

Additional info: Extraneous solutions are common in radical equations due to the process of raising both sides to a power.

Solving Rational Equations

Rational equations involve fractions with polynomials in the numerator and denominator. Solutions must be checked to ensure they do not make any denominator zero.

• Steps to Solve:

  1. Find a common denominator.

  2. Multiply both sides by the common denominator to clear fractions.

  3. Solve the resulting equation.

  4. Check for extraneous solutions (values that make any denominator zero).

• Example: Solve

  • Multiply both sides by :

  • Solve:

  • Check: (no denominator is zero)

Solving Absolute Value Equations

Absolute value equations involve expressions within absolute value bars. The solution process requires considering both the positive and negative cases.

• Steps to Solve:

  1. Set the expression inside the absolute value equal to the positive and negative values of the other side.

  2. Solve each resulting equation.

• Example: Solve

  • Case 1:

  • Case 2:

  • Solutions: or

Additional info: Absolute value equations may have two, one, or no solutions depending on the values involved.