HiSET Math Study Guide: Algebraic Concepts Part 1

Interpreting Parts of an Expression

In algebra, an expression is a combination of numbers, variables, and mathematical operations. Understanding the different parts of an expression is essential for solving problems and analyzing mathematical relationships.

Key Components of an Expression

1. Terms – Individual parts of an expression separated by addition or subtraction.
   • Example: In the expression 3x + 5y – 7, the terms are 3x, 5y, and –7.
2. Factors – Numbers or variables multiplied together within a term.
   • Example: In 4xy, the factors are 4, x, and y because they are multiplied together.
3. Coefficients – The numerical part of a term that multiplies a variable.
   • Example: In 6z, the coefficient is 6 because it multiplies z. If a variable has no visible coefficient, it is understood to be 1 (e.g., the coefficient of x in x + 2 is 1).

Applying These Concepts in Context

Understanding these components helps interpret real-world problems:

• Example 1: Distance Formula
  Suppose the equation d = 60t represents distance (d) in miles traveled in t hours at a constant speed of 60 mph.
  • The term 60t represents the total distance.
  • The coefficient 60 is the speed in mph.
  • The variable t represents the time in hours.
• Example 2: Business Profit
  An expression 5p – 200 models the monthly profit of a business, where p represents the number of products sold.
  • The coefficient 5 means each product contributes $5 to the profit.
  • The term –200 represents a fixed cost or initial expense.

Hint
When analyzing expressions, focus on what each part represents in the given context. Identifying terms, factors, and coefficients makes it easier to understand relationships, solve equations, and apply algebra to real-world situations.

Performing Arithmetic Operations on Polynomials and Rational Expressions

Adding and Subtracting Polynomials

To add or subtract polynomials:

1. Combine like terms, which are terms with the same variable and exponent.
2. Arrange terms in descending order of exponent for clarity.

Example:

$(3x^2 + 5x − 7) + (2x^2 − 4x + 9)$

Combine like terms:

$(3x^2 + 2x^2) + (5x − 4x) + (−7 + 9)$

$5x^2 + x + 2$

For subtraction, distribute the negative sign before combining like terms.

Example:

$(6x^2 + 3x – 8) − (2x^2 − 5x + 4)$

Distribute the negative sign:

$6x^2 + 3x − 8 − 2x^2 + 5x − 4$

Combine like terms:

$(6x^2 − 2x^2) + (3x + 5x) + (-8 − 4)$

$4x^2 + 8x – 12$

Multiplying Polynomials

Multiply each term in one polynomial by each term in the other polynomial and combine like terms.

Using the FOIL Method for Binomials

FOIL stands for:

• First (multiply the first terms)
• Outer (multiply the outer terms)
• Inner (multiply the inner terms)
• Last (multiply the last terms)

$(x + 3)(x − 5)$

1. First: $\, x \cdot x = x^2$
2. Outer: $\, x \cdot (−5) = −5x$
3. Inner: $\, 3 \cdot x = 3x$
4. Last: $\, 3 \cdot (−5) = −15$

Now, combine like terms:

$x^2 − 5x + 3x − 15$

$x^2 − 2x − 15$

If multiplying polynomials with more than two terms, distribute each term individually and then combine like terms.

Dividing Polynomials

Dividing by a Monomial

Divide each term in the numerator separately by the monomial in the denominator.

Example:

$\dfrac{6x^3 + 9x^2 − 12x}{3x}$

Split each term:

$\dfrac{6x^3}{3x} + \dfrac{9x^2}{3x} − \dfrac{12x}{3x}$

Simplify:

$2x^2 + 3x − 4$

Dividing by a Polynomial (Long Division)

Use polynomial long division when dividing by a binomial or larger polynomial.

Example: Divide $x^2 + 3x + 2 \, $ by $ \, x + 1$

1. Divide the first term: $x^2 \div x = x$.
2. Multiply: $x \cdot (x + 1) = x^2 + x$.
3. Subtract: $2(x^2 + 3x + 2) − (x^2 + x) = 2x + 2$
4. Repeat: $2x \div x = 2$, multiply $2(x + 1) = 2x + 2$, subtract to get 0 (no remainder).

Final answer: $ x + 2$.

Operations on Rational Expressions

Rational expressions are fractions with polynomials in the numerator and denominator. Follow fraction rules:

Multiplication and Division

• Multiplication: Multiply straight across and factor if possible.
• Division: Flip the second fraction (reciprocal) and multiply.

Example:

$\dfrac{x^2 − 9}{x + 3} \times \dfrac{x + 2}{x − 3}$

Factor:

$\dfrac{(x − 3)(x + 3)}{x + 3} \times \dfrac{x + 2}{x − 3}$

Cancel common terms:

$\dfrac{\cancel{(x − 3)}(x + 3)}{\cancel{x + 3}} \times \dfrac{x + 2}{\cancel{x − 3}}$

Final answer: $x + 2$

Addition and Subtraction

Find the least common denominator (LCD) before combining terms.

Example:

$\dfrac{3}{x} + \dfrac{5}{x + 2}$

LCD is $x(x + 2)$, so rewrite:

$\dfrac{3(x + 2)}{x(x + 2)} + \dfrac{5x}{x(x + 2)}$

Expand:

$\dfrac{3x + 6}{x(x + 2)} + \dfrac{5x}{x(x + 2)}$

Combine:

$\dfrac{8x + 6}{x(x + 2)}$

Hint:
For FOIL problems, always check if you can factor your final answer to simplify further. For rational expressions, always factor first before multiplying or dividing to cancel common terms.

Algebraic Concepts Part 1 Review Quiz