Algebra: Adding and Subtracting Like Terms

Adding and Subtracting Like Terms

• When adding or subtracting binomials, we need to combine like terms.
• Like terms are terms that have the same variable (e.g., 3x and 4x).
• To add or subtract like terms, we add or subtract the coefficients (numbers in front of the variables).
• Example: 3x + 4x = 7x, and 3x - 4x = -x.

Adding and Subtracting Trinomials

• When adding or subtracting trinomials, we need to combine like terms.
• We can add or subtract like terms by adding or subtracting their coefficients.
• Example: (4x^2 + 3x + 9) + (5x^2 + 7x - 4) = 9x^2 + 10x + 5.

Multiplying Monomials

• When multiplying monomials, we need to multiply the coefficients and add the exponents of the variables.
• Example: x^3 * x^4 = x^7 (because 3 + 4 = 7).
• We can also multiply monomials with different variables, such as x and y.
• Example: x^3 * y^4 = x^3y^4.

Dividing Monomials

• When dividing monomials, we need to divide the coefficients and subtract the exponents of the variables.
• Example: x^9 / x^3 = x^6 (because 9 - 3 = 6).
• We can also divide monomials with different variables, such as x and y.
• Example: x^9y^5 / x^3y^2 = x^6y^3.

Negative Exponents

• When we have a negative exponent, we can move the variable to the other side of the fraction to make it positive.
• Example: x^(-3) = 1/x^3.
• This is useful when we have a negative exponent in the numerator or denominator of a fraction.

Multiplying Monomials with Multiple Variables

• When multiplying monomials with multiple variables, we need to multiply the coefficients and add the exponents of each variable.
• Example: 4x^2y^3 * 7x^3y^4 = 28x^5y^7.

Dividing Monomials with Multiple Variables

• When dividing monomials with multiple variables, we need to divide the coefficients and subtract the exponents of each variable.
• Example: 12x^5y^3 / 4x^2y^2 = 3x^3y^1.

Exponents Raised to Another Exponent

• When we have an exponent raised to another exponent, we need to multiply the two exponents.
• Example: x^3^4 = x^12 (because 3 * 4 = 12).
• This is useful when we have an expression like x^3 raised to the fourth power.

Evaluating Expressions with Exponents

• When we have an expression with exponents, we need to evaluate the exponents first.
• Example: 2^3 * x^4 = 8x^4.
• We can also evaluate expressions with multiple exponents, such as 2^3 * x^4 * y^5.Here are the study notes for the text:

Exponents and Multiplication of Monomials

• To multiply monomials, multiply the coefficients and add the exponents of the same base.
• Example: 2x^2 × 3x^3 = 6x^(2+3) = 6x^5

Multiplying Monomials with Multiplication of Exponents

• When multiplying monomials with exponents, multiply the coefficients and add the exponents of the same base.
• Example: 2x^2 × 3x^3 = 6x^(2+3) = 6x^5
• Note: Exponents with the same base can be added when multiplying.

Multiplying Monomials with Different Bases

• When multiplying monomials with different bases, the coefficients are multiplied, and the bases are combined.
• Example: 2x^2 × 3y^3 = 6x^2y^3

Zero Exponent Rule

• Any number raised to the power of 0 is equal to 1.
• Example: 2^0 = 1, x^0 = 1

Multiplying Binomials

• When multiplying binomials, use the FOIL method:
  • First terms: Multiply the first terms of each binomial.
  • Outer terms: Multiply the outer terms of each binomial.
  • Inner terms: Multiply the inner terms of each binomial.
  • Last terms: Multiply the last terms of each binomial.
• Example: (2x + 3)(x + 4) = 2x^2 + 8x + 6x + 12

Multiplying Binomials by Binomials

• When multiplying binomials by binomials, use the FOIL method and combine like terms.
• Example: (2x + 3)(x + 4) = 2x^2 + 8x + 6x + 12 = 2x^2 + 14x + 12

Multiplying Binomials by Trinomials

• When multiplying binomials by trinomials, use the FOIL method and combine like terms.
• Example: (2x + 3)(x^2 + 4x + 5) = 2x^3 + 11x^2 + 23x + 15

Factoring

• Factoring is the reverse process of multiplying.
• Example: 15x^2 + 24x = 3x(5x + 8)

Factoring by Grouping

• Factoring by grouping is used to factor polynomials with four terms.
• Example: 2x^3 - 6x^2 + 4x - 12 = 2x^2(x - 3) + 4(x - 3) = (2x^2 + 4)(x - 3)

Difference of Perfect Squares

• A difference of perfect squares can be factored as (a + b)(a - b).
• Example: x^2 - 25 = (x + 5)(x - 5)

Adding and Subtracting Like Terms

• Like terms are terms that have the same variable (e.g., 3x and 4x).
• Combine like terms by adding or subtracting their coefficients.
• Example: 3x + 4x = 7x, and 3x - 4x = -x.

Adding and Subtracting Trinomials

• Combine like terms by adding or subtracting their coefficients.
• Example: (4x^2 + 3x + 9) + (5x^2 + 7x - 4) = 9x^2 + 10x + 5.

Multiplying Monomials

• Multiply coefficients and add the exponents of the variables.
• Example: x^3 * x^4 = x^7 (because 3 + 4 = 7).
• Multiply monomials with different variables, such as x and y.
• Example: x^3 * y^4 = x^3y^4.

Dividing Monomials

• Divide coefficients and subtract the exponents of the variables.
• Example: x^9 / x^3 = x^6 (because 9 - 3 = 6).
• Divide monomials with different variables, such as x and y.
• Example: x^9y^5 / x^3y^2 = x^6y^3.

Negative Exponents

• Move the variable to the other side of the fraction to make it positive.
• Example: x^(-3) = 1/x^3.

Multiplying Monomials with Multiple Variables

• Multiply coefficients and add the exponents of each variable.
• Example: 4x^2y^3 * 7x^3y^4 = 28x^5y^7.

Dividing Monomials with Multiple Variables

• Divide coefficients and subtract the exponents of each variable.
• Example: 12x^5y^3 / 4x^2y^2 = 3x^3y^1.

Exponents Raised to Another Exponent

• Multiply the two exponents.
• Example: x^3^4 = x^12 (because 3 * 4 = 12).

Evaluating Expressions with Exponents

• Evaluate the exponents first.
• Example: 2^3 * x^4 = 8x^4.

Exponents and Multiplication of Monomials

• Multiply coefficients and add the exponents of the same base.
• Example: 2x^2 × 3x^3 = 6x^(2+3) = 6x^5.

Multiplying Monomials with Multiplication of Exponents

• Multiply coefficients and add the exponents of the same base.
• Example: 2x^2 × 3x^3 = 6x^(2+3) = 6x^5.

Multiplying Monomials with Different Bases

• Multiply coefficients and combine the bases.
• Example: 2x^2 × 3y^3 = 6x^2y^3.

Zero Exponent Rule

• Any number raised to the power of 0 is equal to 1.
• Example: 2^0 = 1, x^0 = 1.

Multiplying Binomials

• Use the FOIL method: First, Outer, Inner, and Last terms.
• Example: (2x + 3)(x + 4) = 2x^2 + 8x + 6x + 12.

Multiplying Binomials by Binomials

• Use the FOIL method and combine like terms.
• Example: (2x + 3)(x + 4) = 2x^2 + 8x + 6x + 12 = 2x^2 + 14x + 12.

Multiplying Binomials by Trinomials

• Use the FOIL method and combine like terms.
• Example: (2x + 3)(x^2 + 4x + 5) = 2x^3 + 11x^2 + 23x + 15.

Factoring

• Factoring is the reverse process of multiplying.
• Example: 15x^2 + 24x = 3x(5x + 8).