Algebraic Expressions and Evaluation

Evaluating Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operation symbols. To evaluate an expression, substitute the given value for the variable and perform the indicated operations.

• Key Point 1: Substitute the value of the variable into the expression.

• Key Point 2: Follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).

• Example: Evaluate for .

  • Substitute:

  • Calculate:

Exponents and Simplification

Laws of Exponents

The laws of exponents allow us to simplify expressions involving powers. When simplifying, always express the answer with positive exponents.

• Key Point 1: Product of powers:

• Key Point 2: Power of a power:

• Key Point 3: Quotient of powers:

• Key Point 4: Negative exponents:

• Example: Simplify using positive exponents only.

  • Numerator:

  • Denominator:

  • Combine:

Polynomial Operations

Multiplying Binomials

Multiplying binomials can be done using the distributive property or special formulas such as the FOIL method.

• Key Point 1: FOIL stands for First, Outer, Inner, Last terms.

• Key Point 2: Special product formulas:

• Example: Multiply .

  • Apply difference of squares: ,

Factoring by Grouping

Factoring by grouping is a method used to factor polynomials with four terms by grouping pairs of terms and factoring out common factors.

• Key Point 1: Group terms in pairs.

• Key Point 2: Factor out the greatest common factor (GCF) from each group.

• Key Point 3: If a common binomial factor appears, factor it out.

• Example: Factor by grouping.

  • Group:

  • Factor:

  • Final:

Radicals and Rational Expressions

Multiplying Radical Expressions

To multiply radical expressions, use the property and simplify if possible.

• Key Point 1: Combine under a single radical if possible.

• Key Point 2: Simplify the resulting expression.

• Example: Multiply .

  • Expand:

Simplifying Complex Rational Expressions

Complex rational expressions are fractions where the numerator, denominator, or both contain fractions themselves. Simplify by finding a common denominator and reducing.

• Key Point 1: Combine the numerator and denominator into single fractions.

• Key Point 2: Divide the numerator by the denominator (multiply by the reciprocal).

• Example: Simplify .

  • First,

  • So,

  • Now,

  • Since ,

Simplifying Rational Expressions

Adding and Simplifying Rational Expressions

To add rational expressions, find a common denominator, rewrite each fraction, and combine numerators.

• Key Point 1: Factor denominators if possible.

• Key Point 2: Find the least common denominator (LCD).

• Key Point 3: Combine and simplify.

• Example: Simplify .

  • Rewrite as for easier combination.

  • Find LCD and combine numerators.

Mixture Problems

Solving Mixture Problems

Mixture problems involve combining solutions of different concentrations to achieve a desired concentration. Set up an equation based on the amount of pure substance in each solution.

• Key Point 1: Let be the amount of pure water to add.

• Key Point 2: Set up the equation using the concentration formula.

• Key Point 3: Solve for .

• Example: How many liters of pure water should be mixed with a $5 acid to produce a mixture that is water?

  • Original solution: $5 acid ( water)

  • Let = liters of pure water added

  • Total mixture: liters, water content:

  • Set up:

  • Solve for

Summary Table: Key Algebraic Properties