Factorize the following: (a) 15 x^2 y^2+20 x y^3 (b) 14 a^3...

Factorize the following: (a) $15 x^{2} y^{2}+20 x y^{3}$ (b) $14 a^{3} b-12 a^{2} b^{2}$ (c) $2 x^{2}+3 x y-10 x-15 y$ (d) $4 x y-7 y^{2}-12 x+21 y$ (e) $15 x^{2}+8 y+20 x y+6 x$ (f) $6 x y-20+15 x-8 y$ (g) $9 x^{2}+24 x y+16 y^{2}$ (h) $16 x^{2}-40 x y+25 y^{2}$ (i) $25 x^{3} y^{4}-16 x y^{2}$ (j) $(2 x+5 y)^{2}-(x-3 y)^{2}$

Factorize the following:
(a) $15 x^{2} y^{2}+20 x y^{3}$
(b) $14 a^{3} b-12 a^{2} b^{2}$
(c) $2 x^{2}+3 x y-10 x-15 y$
(d) $4 x y-7 y^{2}-12 x+21 y$
(e) $15 x^{2}+8 y+20 x y+6 x$
(f) $6 x y-20+15 x-8 y$
(g) $9 x^{2}+24 x y+16 y^{2}$
(h) $16 x^{2}-40 x y+25 y^{2}$
(i) $25 x^{3} y^{4}-16 x y^{2}$
(j) $(2 x+5 y)^{2}-(x-3 y)^{2}$

Key Concepts

Factoring Quadratic Polynomials

Quadratic factorization involves rewriting a quadratic expression in the form ax^2 + bx + c as the product of two binomials. This is usually achieved by finding two numbers that multiply to ac and add to b, enabling the expression to be broken down into simpler factors.

Greatest Common Factor (GCF) Extraction

This concept involves identifying and factoring out the largest common divisor (numeric or algebraic) shared by all terms in a polynomial. Extracting the GCF simplifies the expression and is often the first step in more advanced factorization techniques.

Factoring by Grouping

This technique is used when a polynomial does not have a common factor across all terms but can be partitioned into groups that each have a common factor. Once common factors are extracted from these groups, a common binomial factor may emerge, allowing for further factorization.

Perfect Square Trinomials

A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. It takes the form a^2 + 2ab + b^2, and recognizing this pattern allows one to factor it immediately into (a + b)^2.

Difference of Squares

This concept refers to a specific type of factorization where an expression of the form a^2 - b^2 can be rewritten as the product (a - b)(a + b). It is a key pattern in algebra that simplifies many seemingly complex expressions.

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