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Questions and Answers
Factor 20x^2 + 25x - 12x - 15 by grouping.
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 ___.
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?
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)
Which statements are true about the polynomial 4x^3 - 6x^2 + 8x - 12? (Select all that apply)
Which polynomial has (3x + 2) as a binomial factor?
Which polynomial has (3x + 2) as a binomial factor?
Which polynomials are prime? (Check all that apply)
Which polynomials are prime? (Check all that apply)
What should Talia do if she cannot find a common factor from her grouping?
What should Talia do if she cannot find a common factor from her grouping?
Factor the polynomial 4x^4 - 20x^2 - 3x^2 + 15 by grouping. What is the resulting expression?
Factor the polynomial 4x^4 - 20x^2 - 3x^2 + 15 by grouping. What is the resulting expression?
Which shows one way to determine the factors of x^3 + 11x^2 - 3x - 33 by grouping?
Which shows one way to determine the factors of x^3 + 11x^2 - 3x - 33 by grouping?
Which shows one way to determine the factors of 12x^3 - 2x^2 + 18x - 3 by grouping?
Which shows one way to determine the factors of 12x^3 - 2x^2 + 18x - 3 by grouping?
Factor -8x^3 - 2x^2 - 12x - 3 by grouping. What is the resulting expression?
Factor -8x^3 - 2x^2 - 12x - 3 by grouping. What is the resulting expression?
Which shows one way to determine the factors of x^3 + 5x^2 - 6x - 30 by grouping?
Which shows one way to determine the factors of x^3 + 5x^2 - 6x - 30 by grouping?
Factor 15x^3 - 5x^2 + 6x - 2 by grouping. What is the resulting expression?
Factor 15x^3 - 5x^2 + 6x - 2 by grouping. What is the resulting expression?
Which shows one way to determine the factors of x^3 + 4x^2 + 5x + 20 by grouping?
Which shows one way to determine the factors of x^3 + 4x^2 + 5x + 20 by grouping?
Which polynomial is prime?
Which polynomial is prime?
Which shows one way to determine the factors of x^3 - 9x^2 + 5x - 45 by grouping?
Which shows one way to determine the factors of x^3 - 9x^2 + 5x - 45 by grouping?
What should Faelyn realize about her work on the polynomial 6x^4 - 8x^2 + 3x^2 + 4?
What should Faelyn realize about her work on the polynomial 6x^4 - 8x^2 + 3x^2 + 4?
Factor 6x^4 - 5x^2 + 12x^2 - 10 by grouping. What is the resulting expression?
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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