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Questions and Answers
What is the result of adding the polynomials (2x^2 + 3x - 1) and (x^2 - 2x - 3)?
What is the result of adding the polynomials (2x^2 + 3x - 1) and (x^2 - 2x - 3)?
- 3x^2 + 5x - 4
- 3x^2 + x - 4 (correct)
- 4x^2 + x - 4
- 3x^2 - 5x - 4
What is the result of multiplying the polynomials (x + 2) and (x + 3)?
What is the result of multiplying the polynomials (x + 2) and (x + 3)?
- x^2 + 4x + 5
- x^2 + 5x + 6 (correct)
- x^2 + 6x + 5
- x^2 + 4x + 6
What is the greatest common factor (GCF) of the terms in the polynomial x^2 + 4x + 4?
What is the greatest common factor (GCF) of the terms in the polynomial x^2 + 4x + 4?
- x + 4
- x + 2 (correct)
- 2
- x + 1
What is the factored form of the polynomial x^2 - 4?
What is the factored form of the polynomial x^2 - 4?
What is the result of subtracting the polynomial (x^2 - 2x - 1) from the polynomial (x^2 + 4x - 2)?
What is the result of subtracting the polynomial (x^2 - 2x - 1) from the polynomial (x^2 + 4x - 2)?
What is the factored form of the polynomial x^2 + 4x + 4?
What is the factored form of the polynomial x^2 + 4x + 4?
What is the result of multiplying the polynomials (x^2 + 2x - 1) and (x - 1)?
What is the result of multiplying the polynomials (x^2 + 2x - 1) and (x - 1)?
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Study Notes
Polynomials
Addition and Subtraction
- To add or subtract polynomials, combine like terms:
- Combine terms with the same variable(s) and coefficient(s)
- Add or subtract coefficients, keeping the variable(s) the same
- Example:
(2x^2 + 3x - 1) + (x^2 - 2x - 3) = 3x^2 + x - 4 - Example:
(x^2 + 4x - 2) - (x^2 - 2x - 1) = 6x + 1
Multiplication
- To multiply polynomials, use the distributive property:
- Multiply each term in one polynomial by each term in the other polynomial
- Combine like terms
- Example:
(x + 2) * (x + 3) = x^2 + 5x + 6 - Example:
(x^2 + 2x - 1) * (x - 1) = x^3 - x^2 - 3x + 1
Factoring
- Factoring polynomials involves expressing them as a product of simpler polynomials:
- Greatest Common Factor (GCF): factor out the largest common factor of all terms
- Difference of Squares:
a^2 - b^2 = (a + b)(a - b) - Sum and Difference:
a^2 + 2ab + b^2 = (a + b)^2anda^2 - 2ab + b^2 = (a - b)^2
- Example:
x^2 + 4x + 4 = (x + 2)^2 - Example:
x^2 - 4 = (x + 2)(x - 2)
Note: These are basic examples and there are many other methods and techniques for factoring polynomials.
Polynomials
Addition and Subtraction
- Combine like terms to add or subtract polynomials
- Combine terms with the same variable(s) and coefficient(s)
- Add or subtract coefficients, keeping the variable(s) the same
- Example:
(2x^2 + 3x - 1) + (x^2 - 2x - 3) = 3x^2 + x - 4 - Example:
(x^2 + 4x - 2) - (x^2 - 2x - 1) = 6x + 1
Multiplication
- Use the distributive property to multiply polynomials
- Multiply each term in one polynomial by each term in the other polynomial
- Combine like terms
- Example:
(x + 2) * (x + 3) = x^2 + 5x + 6 - Example:
(x^2 + 2x - 1) * (x - 1) = x^3 - x^2 - 3x + 1
Factoring
- Factoring polynomials involves expressing them as a product of simpler polynomials
- Greatest Common Factor (GCF): factor out the largest common factor of all terms
- Difference of Squares:
a^2 - b^2 = (a + b)(a - b) - Sum and Difference:
a^2 + 2ab + b^2 = (a + b)^2anda^2 - 2ab + b^2 = (a - b)^2 - Example:
x^2 + 4x + 4 = (x + 2)^2 - Example:
x^2 - 4 = (x + 2)(x - 2)
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