Algebra Fundamentals: Adding Coefficients and Like Terms

Principles of Combining Like Terms

• Definition of Like Terms:

  • Terms in an algebraic expression are defined as like terms when they share identical variable factors or radical components.

  • Algebraic addition is strictly restricted to combining the numerical coefficients of like terms.

• Addition Rule for Coefficients:

  • To combine like terms, add their numerical coefficients together while retaining the shared variable or radical base.

• Single-Variable Linear Addition:

  • Example Expression: $5x + 4x$

  • Analysis: The terms $5x$ and $4x$ are like terms because both share the single linear variable $x$.

  • Coefficients: The coefficients are $5$ and $4$.

  • Step-by-Step Calculation: Add the coefficients $5 + 4 = 9$

  • Final Simplified Result: $9x$

• Multiple-Variable Expressions:

  • Example Expression: $3x + 4y + 5x + 8y$

  • Term Distinction: Terms with differing variables, such as $5x$ and $8y$, are not like terms and cannot be added together.

  • Step 1: Identify and group like terms containing the variable $x$, which are $3x$ and $5x$.

    • Add their coefficients: $3 + 5 = 8$

    • Combined $x$-variable term: $8x$

  • Step 2: Identify and group like terms containing the variable $y$, which are $4y$ and $8y$.

    • Add their coefficients: $4 + 8 = 12$

    • Combined $y$-variable term: $12y$

  • Final Simplified Result: $8x + 12y$

Combining Radical Expressions

• Criteria for Radical Like Terms:

  • Radical terms can only be combined as like terms if they possess the exact same index and radicand.

• Combining Expressions with Multiple Radicands:

  • Example Expression: $3\sqrt{2} + 5\sqrt{7} + 8\sqrt{2} + 3\sqrt{7}$

  • Grouping $\sqrt{2}$ Terms: The terms $3\sqrt{2}$ and $8\sqrt{2}$ both share $\sqrt{2}$.

    • Add coefficients: $3 + 8 = 11$

    • Combined term: $11\sqrt{2}$

  • Grouping $\sqrt{7}$ Terms: The terms $5\sqrt{7}$ and $3\sqrt{7}$ both share $\sqrt{7}$.

    • Add coefficients: $5 + 3 = 8$

    • Combined term: $8\sqrt{7}$

  • Final Simplified Result: $11\sqrt{2} + 8\sqrt{7}$

Combining Polynomial Expressions with Different Exponents

• Exponent Distinction in Like Terms:

  • Terms sharing the same base variable but possessing different exponents (such as $x$ and $x^2$) are not like terms and cannot be directly added together.

• Grouping and Combining Higher-Degree Polynomials:

  • Example Expression: $7x + 4x^2 + 5x + 9x^2$

  • Step 1: Identify and group quadratic terms containing $x^2$, which are $4x^2$ and $9x^2$.

    • Add coefficients: $4 + 9 = 13$

    • Combined quadratic term: $13x^2$

  • Step 2: Identify and group linear terms containing $x$, which are $7x$ and $5x$.

    • Add coefficients: $7 + 5 = 12$

    • Combined linear term: $12x$

  • Final Simplified Result: $13x^2 + 12x$