18 Questions
What is the process of finding the factors of certain given products?
Factorization
Which term is used for products like (a + b)2, (a - b)2, (a + b)(a - b), (a - b)3?
Special products
In factorization of polynomials with integer coefficients, what is the term for a² - b²?
(a + b)(a - b)
What is the result of multiplying (a + b)(a - b)?
$a^2 - b^2$
Which operation can be performed to find the Highest Common Factor (HCF) and Least Common Multiple (LCM) of polynomials?
Factorization
What fundamental operations can be performed on rational algebraic expressions?
Multiplication and Division
Which term describes the process of writing a polynomial as a product of two or more polynomials?
Factorization
In algebra, what do we call the individual polynomials in a product when factorizing a polynomial?
Factors
What type of polynomials are considered factorable in the context of integer coefficients?
Polynomials with integral coefficients
Which of the following represents completely factored polynomials?
(x + 1)(x - 1)
What is the factorization of (4x^2 - 1) when completely factored?
(2x - 1)(2x + 1)
What do we refer to when we say that (x + y) and (x - y) are factors of the polynomial (x^2 - y^2)?
Factors
In the factorization of (16x^4 - 1), which special product is utilized?
(2x - 1)(4x^2 + 1)
What are the factors of 10a - 25?
5 and 2a - 5
Which special product is utilized in the factorization of x^2y^3 + x^3y^2?
x^2y^2
What are the factors of a(b - c)^2 + b(b - c)?
(b - c)(ab - ac + b)
What factors are involved in the factorization of 9x^2 - 16y^2?
(3x)^2 - (4y)^2
What special product is used in the factorization of x^4 - 81y^4?
(x^2 + y^2)(x^2 - y^2)
Study Notes
Special Products and Factorization
- Special products are certain products that occur frequently in algebra, and becoming familiar with them can save time and labor.
- Examples of special products include (a + b)², (a – b)², (a + b)(a – b), and (a – b)³.
Factorization of Polynomials
- Factorization of a polynomial is a process of writing the polynomial as a product of two or more polynomials.
- Each polynomial in the product is called a factor of the given polynomial.
- Factors are restricted to polynomials with integral coefficients.
- A polynomial is said to be completely factored if none of its factors can be further expressed as a product of two polynomials of lower degree and if the integer coefficients have no common factor other than 1 or –1.
Factorization Using Special Products
- Special products can be used to factorize polynomials.
- Examples of factorization using special products include:
- (x + y)² = x² + 2xy + y²
- (x – y)² = x² – 2xy + y²
- (x + y)(x – y) = x² – y²
Factorization by Distributive Property
- Factorization can be done using the distributive property of multiplication over addition.
- Example: 10a – 25 = 5(2a – 5)
- Factors can be found by identifying common terms in the polynomial.
Factorization Involving the Difference of Two Squares
- Difference of two squares can be factorized using the formula (x + y)(x – y) = x² – y².
- Example: 9x² – 16y² = (3x)² – (4y)² = (3x + 4y)(3x – 4y)
- Factors can be found by identifying the difference of two squares in the polynomial.
Test your skills in simplifying and factorizing polynomials with this quiz. Practice simplifying expressions and identifying factors of polynomials. The quiz also covers the concept of factorization of polynomials by expressing them as a product of two or more polynomials.
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