Questions and Answers
What is a polynomial?
A term or a sum of terms whose variables have whole number exponents.
What is a factor?
An integer that divides evenly into another; a polynomial that divides evenly into another; the process of writing a polynomial as a product of prime factors.
What is the factored form of jm + jn + km + kn?
(j + k)(m + n)
What are the factors of ab + 2a + 3b + 6?
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What is the factored form of x² + xy - 2x - 2y?
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What are the factors of mn - 4m - 5n + 20?
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What are the factors of ab + 4 + a + 4b?
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What is the factored form of y² + 10z - 10y - yz?
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What is the factored form of n³ - n² + 3n - 3?
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What are the factors of mr + ns - nr - ms?
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What is the factored form of z³ - 2z² + 9z - 18?
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What are the factors of ab + bc + b² + ac?
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Study Notes
Polynomial Basics
- A polynomial consists of terms or a singular term with whole number exponents on variables.
- Examples of polynomials include x^2 + 3x and 4y^3 - 2y + 1.
Key Definitions
- Factor: An integer or polynomial that divides another evenly. In algebra, factoring involves expressing a polynomial as a product of its prime factors.
- Factoring is crucial for simplifying polynomial expressions and solving equations.
Factoring by Grouping
- Recognize patterns within polynomial expressions to group terms effectively.
- Use the method of factoring by grouping to simplify polynomials into products of binomials.
Example Factorizations
- Factorization of jm + jn + km + kn results in (j + k)(m + n).
- The expression ab + 2a + 3b + 6 can be factored to (a + 3)(b + 2).
- The polynomial x^2 + xy - 2x - 2y simplifies to (x + y)(x - 2).
More Complex Factorizations
- The expression mn - 4m - 5n + 20 factors as (m - 5)(n - 4).
- For ab + 4 + a + 4b, the factored form is (a + 4)(b + 1).
- The polynomial y^2 + 10z - 10y - yz can be factored to (y - z)(y - 10).
Higher Degree Polynomials
- The expression n^3 - n^2 + 3n - 3 simplifies to (n^2 + 3)(n - 1).
- For mr + ns - nr - ms, the factorization results in (m - n)(r - s).
- The polynomial z^3 - 2z^2 + 9z - 18 factors to (z^2 + 9)(z - 2).
Important Notes
- Factors should be listed alphabetically when placing into grids or completing factorization tasks.
- Familiarity with common factorization patterns speeds up the simplification of more complex expressions.
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Description
This quiz covers the essential concepts of polynomials, including key definitions and the method of factoring. Learn how to factor polynomials using techniques like grouping and recognize patterns for simplifying expressions. Test your understanding of these fundamental algebraic skills.
