Questions and Answers
Which binomial is a factor of $9x^2 - 64$?
What are the factors of $m^2 - 6m + 6m - 36$?
Factor $x^3 - 7x^2 - 5x + 35$ by grouping. What is the resulting expression?
(x^2 - 5)(x - 7)
Factor $-7x^3 + 21x^2 + 3x - 9$ by grouping. What is the resulting expression?
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Which term completes the product so that it is the difference of squares? $(−5x−3)(−5x+________)$
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Which monomials are perfect squares? Select three options.
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Which letters from the table represent like terms?
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Adi used algebra tiles to represent the product $(-2x - 1)(2x - 1)$. She used the algebra tiles correctly.
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A square with an area of $A^2$ is enlarged to a square with an area of $25A^2$. How was the side of the smaller square changed?
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Which shows one way to determine the factors of $4x^3 + x^2 - 8x - 2$ by grouping?
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What is the common binomial factor between $2x^3 + 4x$ and $-5x^2 - 10$ after their GCFs have been factored out?
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What is the product of $(-3s + 2t)(4s - t)$?
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Which products result in a difference of squares? Select three options.
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What is the product of $2(x - 4)$?
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What is the value of A from the table after the multiplication of two binomials?
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What is the product of $(6r - 1)(-8r - 3)$?
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Factor $-8x^3 - 2x^2 - 12x - 3$ by grouping. What is the resulting expression?
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The polynomial $10x^3 + 35x^2 - 4x - 14$ is factored by grouping. What is the factored form?
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What is the common factor that is missing from both sets of parentheses: $5x^2() - 2()$?
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What is the square root of $m^6$?
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What is the product of $(2x - 1)(x + 4)$?
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What is the square root of $64y^{16}$?
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What is the square root of $r^{64}$?
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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 $x^3 + 4x^2 + 5x + 20$ by grouping?
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Study Notes
Factoring Polynomials
- Difference of Squares: (9x^2 - 64) can be factored as ((3x - 8)(3x + 8)).
- Grouping Method: For (x^3 - 7x^2 - 5x + 35), factored as ((x^2 - 5)(x - 7)).
- Factoring with Coefficients: (-7x^3 + 21x^2 + 3x - 9) gives factors ((-x + 3)(7x^2 - 3)).
- Common Factor: GCF of (2x^3 + 4x) and (-5x^2 - 10) results in common binomial factor (x^2 + 2) or (2x - 5).
Identifying Terms and Squares
- Perfect Squares: Monomials like (9x^8), (25x^{12}), and (36x^{16}) are perfect squares.
- Square Roots: The square root of (m^6) is (m^3) and for (64y^{16}), it is (8y^8).
Products of Binomials
- Product Calculation: For ((-3s + 2t)(4s - t)), the product is (-12s^2 + 11sr - 2t^2).
- Use of Algebra Tiles: Adi correctly represented the product of ((-2x - 1)(2x - 1)) using algebra tiles.
Additional Factorizations and Products
- Finding Factors: To factor (4x^3 + x^2 - 8x - 2), one way is ((x^2 - 2)(4x + 1)).
- Grouped Products: (5x^2() - 2()) requires the missing common factor of (2x + 7).
Application of Grouping
- Factoring from Polynomial: The polynomial (10x^3 + 35x^2 - 4x - 14) simplifies to ((2x + 7)(5x^2 - 2)).
- Completing the Difference of Squares: ((-5x - 3)(-5x + ____)) is completed with 7.
Other Products
- Calculating Products: From ((2x - 1)(x + 4)), the expanded form is (2x^2 + 7x - 4).
- Binomial Products: Equivalent ways to analyze binomial products and their factors reveal patterns and relationships among coefficients.
Common Roots
- Root Findings: Square roots of (r^{64}) yield (r^{32}) and (r^8) as potential responses.
- Identifying Like Terms: Columns representing binomial multiplications provide clues for like terms.
Summary
- Factoring Techniques: Familiarity with trinomial and binomial products enhances polynomial manipulation skills through techniques like grouping and identifying factors.
- Perfect Squares: Understanding bases and exponents allows students to quickly identify perfect squares and square roots, which is critical for advanced algebra concepts.
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Description
Test your knowledge on factoring polynomials, including the difference of squares, grouping methods, and common factors. This quiz also covers identifying perfect squares and calculating products of binomials using algebraic techniques. Discover your understanding of these essential algebra concepts!
