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Questions and Answers
When adding or subtracting like terms, we need to add or subtract the ______.
coefficients
When multiplying monomials, we need to ______ the coefficients and add the exponents of the variables.
multiply
When dividing monomials, we need to ______ the coefficients and subtract the exponents of the variables.
divide
When we have a ______ exponent, we can move the variable to the other side of the fraction to make it positive.
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When we have an exponent raised to another exponent, we need to ______ the two exponents.
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When multiplying monomials with exponents, we multiply the coefficients and add the ____________________ of the same base.
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In the FOIL method, the first step is to multiply the ____________________ terms of each binomial.
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When multiplying binomials by trinomials, we use the FOIL method and combine like ____________________.
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Any number raised to the power of 0 is equal to ____________________.
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Factoring is the reverse process of ____________________.
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Study Notes
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).
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Description
Learn how to combine like terms when adding or subtracting binomials and trinomials in algebra. Practice adding and subtracting coefficients.
