Algebra Basics: Adding Like Terms

Algebra Basics

• In algebra, you need to learn how to add and subtract like terms.
• A monomial is a single term, while a binomial is a combination of two terms.
• A trinomial is a combination of three terms.

Adding Like Terms

• Like terms are terms that have the same variable raised to the same power.
• To add like terms, you need to add their coefficients.
• For example, 3x + 4x = 7x, and 5 + (-2) = 3.

Adding Two Trinomials

• To add two trinomials, you need to add their like terms.
• For example, (4x^2 + 3x + 9) + (5x^2 + 7x - 4) = (4x^2 + 5x^2) + (3x + 7x) + (9 - 4) = 9x^2 + 10x + 5.

Subtracting Two Trinomials

• To subtract two trinomials, you need to distribute the negative sign to the second trinomial.
• Then, you can add the like terms.
• For example, (5x^2 - 6x - 12) - (7x^2 + 4x - 13) = (5x^2 - 7x^2) + (-6x - 4x) + (-12 + 13) = -2x^2 - 10x + 1.

Multiplying Monomials

• When multiplying monomials, you need to multiply their coefficients and add their exponents.
• For example, x^3 * x^4 = x^(3+4) = x^7.
• x^5 * x^2 = x^(5+2) = x^7.

Dividing Monomials

• When dividing monomials, you need to divide their coefficients and subtract their exponents.
• For example, x^8 / x^3 = x^(8-3) = x^5.
• x^5 / x^7 = x^(5-7) = x^(-2).

Negative Exponents

• When you have a negative exponent, you can move the variable to the other side of the fraction.
• For example, x^(-3) = 1 / x^3.
• 1 / x^(-4) = x^4.

Multiplying Monomials with Negative Exponents

• When multiplying monomials with negative exponents, you need to multiply their coefficients and add their exponents.
• For example, x^(-3) * x^4 = x^(-3+4) = x^1.

Dividing Monomials with Negative Exponents

• When dividing monomials with negative exponents, you need to divide their coefficients and subtract their exponents.
• For example, x^(-5) / x^(-3) = x^(-5-(-3)) = x^(-2).

Exponents Raised to Another Exponent

• When you have an exponent raised to another exponent, you need to multiply the exponents.
• For example, (x^3)^4 = x^(3*4) = x^12.
• (x^4)^6 = x^(4*6) = x^24.

Multiplying and Dividing Numbers with Exponents

• When multiplying and dividing numbers with exponents, you need to follow the order of operations.
• For example, 2^3 * x^4 = 8x^4.
• 2^3 / x^3 = 8/x^3.### Exponent Laws and Simplification
• Monomials can be raised to a power; apply the exponent to each factor inside the monomial.
• In multiplication of exponents (e.g., ( x^a \times x^b = x^{a+b} )), add the exponents.
• Any expression raised to the zero power equals one (e.g., ( a^0 = 1 )).
• Negative signs outside versus inside parentheses affect the final answer differently during exponentiation.

Multiplication of Polynomials

• A monomial multiplied by a binomial involves distributing the monomial to each term in the binomial.
• Example: ( 3x(5x + 8) ) results in ( 15x^2 + 24x ).
• Multiplying a monomial by a trinomial doubles the number of initial terms, resulting in three products to combine.

FOIL Method for Binomials

• FOIL (First, Outer, Inner, Last) is used to multiply two binomials, resulting in four terms.
• Example: ( (2x + 3)(x - 2) ) yields ( 2x^2 - 4x + 3x - 6 ), simplifying to ( 2x^2 - x - 6 ).

Factoring Polynomials

• Factoring is the reverse process of multiplication.
• To factor, identify the greatest common factor (GCF) and factor it out.
• Example: From ( 8x + 12 ), the GCF is 4, resulting in ( 4(2x + 3) ).
• Recognizing patterns such as differences of squares can simplify factoring (e.g., ( x^2 - 25 = (x + 5)(x - 5) )).

Difference of Squares

• The difference of squares formula is ( a^2 - b^2 = (a + b)(a - b) ).
• Applies to expressions like ( x^2 - 9 = (x + 3)(x - 3) ) and ( 4x^2 - 25 = (2x + 5)(2x - 5) ).

Grouping in Factoring

• Factor by grouping is useful for polynomials with four terms sharing a proportional relationship in their coefficients.
• Group terms, factor out the GCF from each group, and recognize the common binomial factors.

Example Problems

• For ( 2x^3 - 6x^2 + 4x - 12 ): Factor out GCF from each group to reveal factors.
• Confirm factoring accuracy by multiplying the factors back to check consistency with the original expression.

Other Key Facts

• The product of a polynomial can be thoroughly simplified using distribution, combining like terms, and ensuring that all products follow exponent rules.
• When multiple terms are produced through multiplication, thorough organization helps in correctly identifying and combining like terms to simplify the output.

Practice

• Encourage attempting similar problems to solidify understanding of factoring, multiplying, and applying exponent rules as fundamental skills in algebra.

Algebra Basics

• Monomials consist of a single term, binomials have two terms, and trinomials contain three terms.

Adding Like Terms

• Like terms share the same variable raised to the same power.
• Combine coefficients to add them; for example, 3x + 4x = 7x.

Adding Two Trinomials

• To add two trinomials, combine their like terms carefully across each variable.
• Example: (4x² + 3x + 9) + (5x² + 7x - 4) results in 9x² + 10x + 5.

Subtracting Two Trinomials

• Distribute the negative sign to the second trinomial before combining like terms.
• Example: (5x² - 6x - 12) - (7x² + 4x - 13) simplifies to -2x² - 10x + 1.

Multiplying Monomials

• Multiply coefficients and add exponents when multiplying monomials.
• For example, x³ * x⁴ = x^(3+4) = x⁷.

Dividing Monomials

• Divide coefficients and subtract exponents when dividing monomials.
• Example: x⁸ / x³ = x^(8-3) = x⁵.

Negative Exponents

• A negative exponent indicates the reciprocal of the base; x^(-n) = 1/x^n.
• Example: x^(-3) = 1/x³.

Multiplying Monomials with Negative Exponents

• Apply the same rules: multiply coefficients and add exponents.
• Example: x^(-3) * x⁴ = x^1.

Dividing Monomials with Negative Exponents

• Divide coefficients and subtract exponents, even with negatives involved.
• Example: x^(-5) / x^(-3) = x^(-2).

Exponents Raised to Another Exponent

• When raising an exponent to another exponent, multiply the exponents.
• Example: (x³)⁴ = x^(3*4) = x¹².

Multiplying and Dividing Numbers with Exponents

• Follow the order of operations while handling exponents in expressions.
• Example: 2³ * x⁴ = 8x⁴ and 2³ / x³ = 8/x³.

Exponent Laws and Simplification

• Apply exponents to each factor of a monomial when raised to a power.
• The zero exponent law states a^0 = 1, regardless of a's value.
• Pay attention to the placement of negative signs during exponentiation.

Multiplication of Polynomials

• Distribute monomials across binomials to achieve multiple products.
• Example: 3x(5x + 8) expands to 15x² + 24x.

FOIL Method for Binomials

• FOIL (First, Outer, Inner, Last) method generates four terms from multiplying two binomials.
• Example: (2x + 3)(x - 2) results in 2x² - x - 6.

Factoring Polynomials

• Factoring simplifies polynomials by identifying and removing their greatest common factors (GCF).
• Example: GCF of 8x + 12 is 4, yielding 4(2x + 3).

Difference of Squares

• The difference of squares formula states a² - b² = (a + b)(a - b).
• Useful for expressions such as x² - 9 and 4x² - 25.

Grouping in Factoring

• Effective for polynomials with four terms by grouping pairs and factoring out the GCF from each.

This quiz covers the fundamental concepts of algebra, focusing on adding and subtracting like terms, monomials, binomials, and trinomials. Understand how to identify and combine like terms with examples to enhance your algebra skills.