Questions and Answers
Which monomials are perfect squares? (Select all that apply)
What are the two binomial factors for $x^2 - 25$?
(x - 5)(x + 5)
What is the value of $ac$ for the polynomial $9x^2 - 16$?
-144
What is the value of $b$ for the polynomial $9x^2 - 16$?
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What two numbers multiply to get $ac$ and add to get $b$ for the polynomial $9x^2 - 16$?
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What is the factored form of $9x^2 - 16$?
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What is the factored form of $36x^4 - 25$?
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Which expressions represent a perfect square monomial and its square root? (Select all that apply)
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Complete the factorization of $4x^2 - 9$.
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What are the factors of $x^2 - 144$?
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What is the value of $ac$ for the polynomial $16x^2 - 1$?
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What is the value of $b$ for the polynomial $16x^2 - 1$?
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What two numbers multiply to get $ac$ and add to get $b$ for the polynomial $16x^2 - 1$?
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What is the factored form of $16x^2 - 1$?
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What are the factors of $36x^8 - 49$?
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Which expression represents the area of the remaining paper shape in square centimeters after a corner of 16 square centimeters is removed?
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Analyze Cecile's work. Is it correct?
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What is the factorization of $121b^4 - 49$?
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Which polynomial can be simplified to a difference of squares?
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What is the factorization of $49b^2 - 81$?
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Which expression is equivalent to $100 n^2 - 1$?
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The area of a square can be represented by the expression $x^{10}$. What monomial represents a side of the square?
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Which expression is equivalent to $36x^2 - 25$?
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Which expressions are equivalent to $25x^4 - 64$? (Select three options)
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Which term is a perfect square of the root $3x^4$?
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What is the square root of $r^{64}$?
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What are the factors of $x^2 - 100$?
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Study Notes
Perfect Squares and Monomials
- Perfect square monomials include: 1, 49x², and 81x⁶.
- Expressions like 121;11, 4x²;2x, and 49x⁴;7x² represent perfect squares and their square roots.
Difference of Squares
- For x² - 25, the binomial factors are (x - 5)(x + 5).
- The equation 9x² - 16 factors to (3x - 4)(3x + 4).
- Value of ac = -144
- Value of b = 0
- Numbers that multiply to ac and add to b are -12 and 12.
- 36x² - 25 factors to (6x - 5)(6x + 5).
- The expression 49b² - 81 factors to (7b - 9)(7b + 9).
- The factors of x² - 144 are (x - 12)(x + 12).
- 121b⁴ - 49 factors to (11b² + 7)(11b² - 7).
Factorization Techniques
- Use a² - b² = (a + b)(a - b) for factoring difference of squares.
- For 36x⁸ - 49, the factors are (6x⁴ - 7)(6x⁴ + 7).
- Factor 4x² - 9 as (2x - 3)(2x + 3).
Expressions and Areas
- Area of a square can be expressed as x¹⁰; its side length is x⁵.
- Remaining area after removing a corner with area 16 cm² from a square area of 9x² is represented by (3x - 4)(3x + 4).
Equations and Simplifications
- The expression equivalent to 100n² - 1 is (10n)² - (1)².
- 36x² - 25 can be simplified to (6x)² - (5)².
- To simplify 25x⁴ - 64, three equivalent expressions include:
- 25x⁴ + 40x - 40x - 64
- 25x⁴ + 13x - 13x - 64
- (5x² + 8)(5x² - 8).
Other Key Points
- Perfect square corresponding to the root 3x⁴ is 9x⁸.
- The square root of r⁶⁴ is r⁸.
- Factorization for x² - 100 yields (x - 10)(x + 10).
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Description
Test your knowledge on factoring polynomials with this quiz focused on perfect squares and the difference of squares. You'll answer questions about identifying monomials and factoring binomials, enhancing your understanding of quadratic expressions.
