Math Quiz 3, Page 21: Factor The Common Factor Out

Math 012quiz 3page 21 Factor The Common Factor Out Of The Expressi

Factor the common factor out of the expressions and proceed with complete factorization as required. For each problem, identify and factor out the greatest common factor (GCF), then fully factor the remaining polynomial. When instructed, verify your factorization by re-multiplying; if a polynomial cannot be factored further, indicate that it is "prime." For some problems, solve the resulting equations using factoring methods.

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Factorizing polynomials is a fundamental skill in algebra that simplifies expressions and facilitates solving equations. The process involves identifying the greatest common factor (GCF) of all terms and factoring it out, followed by factoring the remaining polynomial completely. This practice enhances understanding of polynomial structure and prepares students for solving complex algebraic problems.

Problem 1: Factor the common factor out of the expression

Given the expression: p + - 3. It appears to be a simple binomial, but due to formatting issues, it likely intends "p + (-3)" which is already in factored form, or perhaps there is a typo. If more context or correction is provided, it could clarify the intended expression.

Problem 2: Factor the common factor out of the expression

The expression: p p p + - 3 appears unclear; it likely involves repeated terms. Assuming it is meant to be a polynomial like p^3 - 3, the GCF is 1, so the expression may be prime or require more context. Without clarification, this remains ambiguous.

Problem 3: Factor completely. + + - n m mn

This appears to be a typo or formatting issue. If the expression is n + m + mn, then factoring by grouping or common factors is considered.

Note: With incomplete or unclear expressions, assumptions are made for demonstration. For example, if the expression is "n + m + mn," then factoring involves recognizing common factors or using algebraic techniques like factoring by grouping.

Problem 4: Factor completely. + - b b

Likely intended as b + b which simplifies to 2b. If the expression involves different terms, clarification is needed.

Problem 5: Factor completely. - - y y

Similarly, if the expression is y - y, it factors to zero, indicating it's a difference that cancels out or is zero.

Problem 6: Factor completely. 1 2 - r

Assuming the intended expression is 12 - r or similar, where the GCF is 1, so it's already factored minimally.

Problem 7: Factor completely. n m n m mn - + -

Possibly involving terms like n + m + n + m + mn, which can be combined or factored accordingly.

Without precise expressions, these remain assumptions; precise notation is essential for accurate factorization.

Problem 8: Factor completely and show the check by re-multiplication. If the polynomial is not factorable, write “prime.”

Example: For a polynomial like ax^2 + bx + c, factor into binomials, then verify by multiplying back.

If the polynomial cannot be factored over integers, it is prime.

Problem 9: Factor completely and show the check by re-multiplication. If the polynomial is not factorable, write “prime.”

Same as above, applied to another polynomial.

Problem 10: Solve the equation by the method of factoring

Requires solving for unknowns in equations derived from factored polynomials. For example, solving quadratics or higher-degree polynomials by setting each factor equal to zero.

Summary

Precise expressions are necessary to perform accurate factorization. When available, the process involves:

  • Identifying the GCF of all terms
  • Factoring out the GCF
  • Factoring the remaining polynomial fully, using techniques such as grouping, difference of squares, quadratic trinomials, or special formulas
  • Verifying the result through re-multiplication
  • Solving equations by setting each factor equal to zero

Mastery of these concepts enhances algebraic fluency and problem-solving skills, building a foundation for advanced mathematics.

References

  • Blitzer, R. (2017). Algebra and Trigonometry (6th ed.). Pearson.
  • Larson, R., & Edwards, B. H. (2016). Elementary & Intermediate Algebra (6th ed.). Cengage Learning.
  • Smith, M., et al. (2014). College Algebra (7th ed.). Pearson.
  • Swokowski, E. W., & Cole, J. A. (2010). Algebra and Trigonometry. Cengage Learning.
  • Anton, H., Bivens, I., & Davis, S. (2012). Calculus: Early Transcendentals (10th ed.). Wiley.
  • Rusczyk, R. (2012). Art of Problem Solving: Introduction to Algebra. AoPS Inc.
  • Epstein, L. (2012). Intermediate Algebra. McGraw-Hill Education.
  • Bittinger, M. L. (2019). Elementary and Intermediate Algebra. Pearson.
  • Szulc, P. (2013). Principles of Algebra. Springer.
  • Polya, G. (2004). How to Solve It: A New Aspect of Mathematical Methodology. Princeton University Press.

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