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Questions and Answers
What is the complete factored form of $10 + 270y^3$?
What is the complete factored form of $10 + 270y^3$?
- 10(1 - 3y)(1 - 3y + 9y^2)
- 10(1 + 3y)(1 - 3y + 9y^2) (correct)
- 10(1 + 3y)(1 + 3y + 9y^2)
- 10(1 - 3y)(1 + 3y + 9y^2)
What is the other side of a rectangular garden with area $9t - 64$ square units and one side $3t - 8$?
What is the other side of a rectangular garden with area $9t - 64$ square units and one side $3t - 8$?
- 3t - 8
- 3t + 8
- t - 8
- t + 8 (correct)
Which expression represents the factored form of $27a^3b - 9a^3b$?
Which expression represents the factored form of $27a^3b - 9a^3b$?
- 9a^3b(3 - 1)
- 9a^3b(3(3 - 1)) (correct)
- 9a^3b(3^2 - 1)
- 9a^3b(3^2 + 3)
What is the correct factored expression of $x^2 - 121y$?
What is the correct factored expression of $x^2 - 121y$?
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What is the factored form of $x^3 - 27$?
What is the factored form of $x^3 - 27$?
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Which expression is a common factor in $12x - 27x$?
Which expression is a common factor in $12x - 27x$?
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When completely factored, what form does $128 - 200m$ take?
When completely factored, what form does $128 - 200m$ take?
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What is the factored form of $4x^2 - 64$?
What is the factored form of $4x^2 - 64$?
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Which is the factored form of $8h + 27j$?
Which is the factored form of $8h + 27j$?
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Study Notes
Factoring Polynomials
- Factoring helps to simplify expressions and solve equations by breaking them down into their components.
- Common scenarios for factoring include the difference of two squares, sum and difference of cubes, and using the greatest common factor (GCF).
Difference of Two Squares
- Formula: ( a^2 - b^2 = (a - b)(a + b) ).
- Can be used with polynomials, e.g., ( x^2 - 81 = (x - 9)(x + 9) ).
Patterns in Factoring
- Identification of patterns aids in factoring effectively:
- Example 1: ( 4x^2 - 49 = (2x - 7)(2x + 7) ).
- Example 2: ( 16x^2 - 81y^2 = (4x - 9y)(4x + 9y) ).
Common Monomial Factor
- The process begins by identifying the GCF of all terms:
- Example: For ( 3x^2 - 12y^2 ), the GCF is ( 3 ), so it factors to ( 3(x^2 - 4y^2) ).
Polynomial Expressions
- Certain binomials remain factorable after initial factoring:
- ( 1 - 16x^8 = (1 + 4x^4)(1 - 4x^4) ), where ( (1 - 4x^4) ) can further reduce.
Greatest Common Factor (GCF)
- Important for simplifying expressions or finding factors:
- GCF of 24 and 54 is 6, while GCF of 20, 24, and 40 is 4.
- GCF of monomials can often be represented as the lowest power of common variables.
Factorization Examples
- ( 7x - 7 = 7(x - 1) ).
- Area of a rectangle problem: Given area ( 9t - 64 ) and one side ( 3t - 8 ), the other side can be found using division.
Additional Activities
- Encourage practice to factor expressions completely:
- ( 27ab - 9ab = 9ab(3 - 1) ).
- ( x - 121y = (x - 11y)(x + 11y) ) demonstrates using difference of squares.
Learning Objectives
- Master patterns in factoring polynomials.
- Factor completely using greatest common monomial factors.
- Understand and apply techniques for the difference of squares and sum/difference of cubes.
Self-Assessment
- Questions to check understanding of GCF and factoring processes.
- Engage with multiple-choice questions and apply knowledge practically through examples.
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