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Questions and Answers
Which monomials are perfect squares? (Select all that apply)
Which monomials are perfect squares? (Select all that apply)
- 1 (correct)
- 49x^2 (correct)
- 81x^6 (correct)
- None of the above
What are the two binomial factors for $x^2 - 25$?
What are the two binomial factors for $x^2 - 25$?
(x - 5)(x + 5)
What is the value of $ac$ for the polynomial $9x^2 - 16$?
What is the value of $ac$ for the polynomial $9x^2 - 16$?
-144
What is the value of $b$ for the polynomial $9x^2 - 16$?
What is the value of $b$ for the polynomial $9x^2 - 16$?
What two numbers multiply to get $ac$ and add to get $b$ for the polynomial $9x^2 - 16$?
What two numbers multiply to get $ac$ and add to get $b$ for the polynomial $9x^2 - 16$?
What is the factored form of $9x^2 - 16$?
What is the factored form of $9x^2 - 16$?
What is the factored form of $36x^4 - 25$?
What is the factored form of $36x^4 - 25$?
Which expressions represent a perfect square monomial and its square root? (Select all that apply)
Which expressions represent a perfect square monomial and its square root? (Select all that apply)
Complete the factorization of $4x^2 - 9$.
Complete the factorization of $4x^2 - 9$.
What are the factors of $x^2 - 144$?
What are the factors of $x^2 - 144$?
What is the value of $ac$ for the polynomial $16x^2 - 1$?
What is the value of $ac$ for the polynomial $16x^2 - 1$?
What is the value of $b$ for the polynomial $16x^2 - 1$?
What is the value of $b$ for the polynomial $16x^2 - 1$?
What two numbers multiply to get $ac$ and add to get $b$ for the polynomial $16x^2 - 1$?
What two numbers multiply to get $ac$ and add to get $b$ for the polynomial $16x^2 - 1$?
What is the factored form of $16x^2 - 1$?
What is the factored form of $16x^2 - 1$?
What are the factors of $36x^8 - 49$?
What are the factors of $36x^8 - 49$?
Which expression represents the area of the remaining paper shape in square centimeters after a corner of 16 square centimeters is removed?
Which expression represents the area of the remaining paper shape in square centimeters after a corner of 16 square centimeters is removed?
Analyze Cecile's work. Is it correct?
Analyze Cecile's work. Is it correct?
What is the factorization of $121b^4 - 49$?
What is the factorization of $121b^4 - 49$?
Which polynomial can be simplified to a difference of squares?
Which polynomial can be simplified to a difference of squares?
What is the factorization of $49b^2 - 81$?
What is the factorization of $49b^2 - 81$?
Which expression is equivalent to $100 n^2 - 1$?
Which expression is equivalent to $100 n^2 - 1$?
The area of a square can be represented by the expression $x^{10}$. What monomial represents a side of the square?
The area of a square can be represented by the expression $x^{10}$. What monomial represents a side of the square?
Which expression is equivalent to $36x^2 - 25$?
Which expression is equivalent to $36x^2 - 25$?
Which expressions are equivalent to $25x^4 - 64$? (Select three options)
Which expressions are equivalent to $25x^4 - 64$? (Select three options)
Which term is a perfect square of the root $3x^4$?
Which term is a perfect square of the root $3x^4$?
What is the square root of $r^{64}$?
What is the square root of $r^{64}$?
What are the factors of $x^2 - 100$?
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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