Questions and Answers
Factor 20x^2 + 25x - 12x - 15 by grouping.
(5x - 3)(4x + 5)
The GCF of the group (14x^2 - 7x) is ___ and of the group (6x - 3) is ___.
7x, 3
What is the factored form of the polynomial 3x^4 - 2x^2 + 15x^2 - 10?
(x^2 + 5)(3x^2 - 2)
Which statements are true about the polynomial 4x^3 - 6x^2 + 8x - 12? (Select all that apply)
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Which polynomial has (3x + 2) as a binomial factor?
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Which polynomials are prime? (Check all that apply)
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What should Talia do if she cannot find a common factor from her grouping?
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Factor the polynomial 4x^4 - 20x^2 - 3x^2 + 15 by grouping. What is the resulting expression?
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Which shows one way to determine the factors of x^3 + 11x^2 - 3x - 33 by grouping?
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Which shows one way to determine the factors of 12x^3 - 2x^2 + 18x - 3 by grouping?
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Factor -8x^3 - 2x^2 - 12x - 3 by grouping. What is the resulting expression?
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Which shows one way to determine the factors of x^3 + 5x^2 - 6x - 30 by grouping?
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Factor 15x^3 - 5x^2 + 6x - 2 by grouping. What is the resulting expression?
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Which shows one way to determine the factors of x^3 + 4x^2 + 5x + 20 by grouping?
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Which polynomial is prime?
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Which shows one way to determine the factors of x^3 - 9x^2 + 5x - 45 by grouping?
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What should Faelyn realize about her work on the polynomial 6x^4 - 8x^2 + 3x^2 + 4?
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Factor 6x^4 - 5x^2 + 12x^2 - 10 by grouping. What is the resulting expression?
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Study Notes
Factoring Polynomials by Grouping
- To factor polynomials using grouping, rearrange and group terms with common factors, then extract the greatest common factor (GCF) from each group.
- For example, in the polynomial 20x² + 25x - 12x - 15, grouping yields (20x² - 12x) + (25x - 15), leading to the factors of (5x - 3)(4x + 5).
Finding GCF and Common Factors
- Identify the GCF of grouped terms; for 14x² + 6x - 7x - 3, the GCFs are 7x from the first group and 3 from the second group, resulting in the factored expression (7x + 3)(2x - 1).
- Evaluate whether a polynomial can be regrouped and factored in multiple ways, such as in the example of 4x³ - 6x² + 8x - 12.
Recognizing Prime Polynomials
- A polynomial is classified as prime if it cannot be factored further into integers or simpler polynomials.
- Examples of prime polynomials include 15x² + 10x - 9x + 7 and 9x³ + 11x² + 3x - 33.
Steps in Grouping and Factoring
- Group terms strategically, as shown in polynomial examples; for instance, 15x² - 3x - 20x + 4 can be grouped as 3x(5x - 1) + 4(-5x + 1), necessitating adjustment for negative factors to create matching binomials.
- Apply grouping techniques, such as those in 12x³ - 2x² + 18x - 3, which demonstrates the common factor 2x²(6x - 1) + 3(6x - 1).
Factored Expressions and Results
- The resulting expressions from various polynomials when factored include (x² + 5)(3x² - 2), (4x² - 3)(x² - 5), and (5x² + 2)(3x - 1).
- Specific grouping techniques yield factors like x²(x + 5) - 6(x + 5) and x(x² + 5) - 9(x² + 5).
Analyzing Polynomial Characteristics
- Determining if a polynomial is prime involves evaluating the possibility of further factorization; with Faelyn's observation, her polynomial was determined to be prime.
- If common factors do not emerge from grouping, reevaluating the terms and their relationships can clarify their characteristics, potentially revealing prime status.
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Description
Test your knowledge of factoring polynomials using the double grouping method with this comprehensive quiz. You'll learn how to identify common factors and combine terms to factor quadratic expressions effectively. Perfect for Algebra 1 students looking to strengthen their skills in polynomial factoring.
