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Questions and Answers
What is the definition of the expression $x^2 - 4$?
What is the definition of the expression $x^2 - 4$?
- perfect square
- difference of squares (correct)
- binomial product
- sum of squares
What is the factorization of $x^2 + 2x - 8$?
What is the factorization of $x^2 + 2x - 8$?
factor directly
What is the factorization of $v^2n - 2v^2 - 4n + 8$?
What is the factorization of $v^2n - 2v^2 - 4n + 8$?
(n-2)(v^2-4)
What are the two numbers that multiply to -30 and add to 13?
What are the two numbers that multiply to -30 and add to 13?
What method is used in the problem $4x^2 - 25$?
What method is used in the problem $4x^2 - 25$?
What is the factorization of $x^2 + 12x + 35$?
What is the factorization of $x^2 + 12x + 35$?
What is the factorization of the trinomial $3x^2 + 7x - 20$ using MARFF?
What is the factorization of the trinomial $3x^2 + 7x - 20$ using MARFF?
What is the factorization of $6a^3 - 3a^2 + 8a - 4$ by grouping?
What is the factorization of $6a^3 - 3a^2 + 8a - 4$ by grouping?
What is the factorization of $4p^6 - 49$ using the difference of squares method?
What is the factorization of $4p^6 - 49$ using the difference of squares method?
Is the expression $x^2 + 23x + 11$ factorable?
Is the expression $x^2 + 23x + 11$ factorable?
What is the factorization of $4x^2 - 25$?
What is the factorization of $4x^2 - 25$?
What is the factorization of $x^2 + 16x - 105$?
What is the factorization of $x^2 + 16x - 105$?
What is the factorization of $4x^3 + 28x^2 - 3x - 21$?
What is the factorization of $4x^3 + 28x^2 - 3x - 21$?
What is the simplified form of the polynomial $(3x + 7) + (3x + 7) + (2x - 3) + (2x - 3)$?
What is the simplified form of the polynomial $(3x + 7) + (3x + 7) + (2x - 3) + (2x - 3)$?
What is the area of a rectangle with sides $(2x - 3)$ and $(3x + 7)$?
What is the area of a rectangle with sides $(2x - 3)$ and $(3x + 7)$?
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Study Notes
Factoring Types and Techniques
- Difference of Squares: Expresses as (a^2 - b^2 = (a - b)(a + b)). Example: (x^2 - 4) factors to ((x - 2)(x + 2)).
- Trinomial Factoring: Directly factor polynomials of the form (x^2 + bx + c). Example: (x^2 + 12x + 35) factors to ((x + 7)(x + 5)).
- Grouping Method: Used for polynomials with four terms, grouping pairs for factoring. Example: (6a^3 - 3a^2 + 8a - 4) factors to ((2a - 1)(3a^2 + 4)).
Solving Polynomials
- Factoring Trinomials: Combine two numbers that multiply to (c) and add to (b). Example: For (x^2 + 2x - 8), factors into ((x + 4)(x - 2)).
- Polynomials with Multiple Methods: Different methods may suit various expressions. Example: (4x^2 - 25) can be solved using the difference of squares to give ((2x - 5)(2x + 5)).
Number Properties
- Numbers that multiply to give a negative product and add up to a positive can include one positive and one negative. Example: The numbers for -30 that add to 13 are 15 and -2.
Factorability
- Some polynomials cannot be factored using integers. Example: (x^2 + 23x + 11) is not factorable with rational coefficients.
- Completely factorable forms include products of simpler binomials or trinomial expressions. Example: (x^2 + 16x - 105) factors into ((x + 21)(x - 5)).
Polynomial Simplification
- Simplicity is key for polynomial expressions. Example: Adding ((3x + 7)) twice and ((2x - 3)) twice simplifies to (10x + 8).
- When finding areas of geometric shapes, expand product forms. Example: The area of a rectangle with sides ((2x - 3)) and ((3x + 7)) simplifies to (6x^2 + 5x - 21).
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