Questions and Answers
What is the factorization of the expression $8x^3 + y^2 + 27z^2 - 18xyz$?
What is the expanded form of $(2x + 1)^3$?
Which identity can be used to evaluate the product $(x + 4)(x + 10)$?
Which expression correctly factors $4x^4 + 5x^2 + 1$?
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What is the result of the product $95 imes 96$ evaluated without direct multiplication?
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What is the result when verifying $(x + y)^3 = (x + y)(x^2 - xy + y^2)$?
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What is the factorization of $27 - 125 - 135a + 225a^2$?
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Which expression is the correct factorization for $9x^2 + 6xy + y^2$?
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What type of expression is $100 - x^2$ and how can it be factored?
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What expression represents the area of a rectangle if the area is given as $25a^2 - 35a + 12$?
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Study Notes
Factorisation and Algebraic Identities
- Factorisation example: ( 8x^3 + y^2 + 27z^2 - 18xyz ) can be factored as ( (2x+y+3z)(4x^2 + y^2 + 9z^2 - 2xy - 3yz - 6xz) ).
Products Using Identities
- Use algebraic identities to find products:
- ( (x+4)(x+10) = x^2 + 14x + 40 )
- ( (x+8)(x-10) = x^2 - 2x - 80 )
- ( (3x+4)(3x-4) = 9x^2 - 16 )
- ( (x+\frac{4}{3})(x-\frac{4}{3}) = x^2 - \left(\frac{4}{3}\right)^2 )
- ( (3-2x)(3+2x) = 9 - 4x^2 )
Evaluating Products Without Direct Multiplication
- ( 103 \times 107 = 11051 ) (using ( (100 + 3)(100 + 7) = 100^2 + 10 + 21 ))
- ( 95 \times 96 = 9120 ) (using ( (100 - 5)(100 - 4) = 10000 - 500 - 400 ))
- ( 104 \times 96 = 9984 ) (using ( (100 + 4)(100 - 4) = 10000 - 16 ))
Factorisation Examples
- ( 9x^2 + 6xy + y^2 = \left(3x+y\right)^2 )
- ( 4y^2 - 4y + 1 = (2y-1)^2 )
- ( 100 - x^2 = (10-x)(10+x) )
Expanding Using Identities
- Expand ( (1 - 2x + 3x^2)^2 ) and ( (x^3 + 2x + 1)^2 ) using suitable binomial expansion identities.
Advanced Factorisation
- Factor ( 4x^4 + 5x^2 + 1 ), ( 4x^4 + 8x^3 + 24x^2 + 32x + 16 ), ( 4x^4 - 8x^3 - 2x^2 + 2x - 1 ) through trial or polynomial long division.
Evaluation of Cubes
- Expand ( (2x + 1)^3 ) to ( 8x^3 + 12x^2 + 6x + 1 ).
Factorisation of Higher Orders
- Factor ( 8x^3 + b^3 + 12x^2b + 6ab^2 ) and ( 27y^3 + 125 ) to reveal their component structures.
Verification of Identities
- Verify ( (x + y)^3 = (x + y)(x^2 - xy + y^2) ) and similarly for ( (x - y)^3 ).
Factorisation of Polynomials
- Apply the Factor Theorem to determine factors of polynomials like ( x^3 + 6x^2 + 11x + 6 ).
Length of Pendulum
- The formula for the length of a pendulum ( l ) is given by ( l = \frac{gT^2}{4\pi^2} ).
Practical Example
- For a pendulum with a period of 2 seconds, with gravitational acceleration ( g = 9.8 ), the length is approximately ( 0.994 ) meters.
Algebraic Identities Summary
- Key identities involving cubes:
- ( x^3 + y^3 + z^3 - 3xyz = (x+y+z)(x^2+y^2+z^2 - xy - xz - yz) )
Employing Factor Theorem
- Determine whether a linear polynomial is a factor of a given cubic polynomial by substituting its roots and checking for zero remainders.
Conclusion
- A thorough understanding of algebraic identities, factorisation techniques, and methods for evaluating and reformatting polynomial expressions forms the cornerstone of mastery in polynomial algebra.
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Description
This quiz covers factorisation techniques and the application of algebraic identities in multiplying binomials and evaluating products without direct multiplication. You will explore various examples, understand identities, and learn to expand quadratic expressions. Test your skills in algebra with these engaging questions!
