Factorize the following: (a) 36 x^3 y^2-8 x^2 y (e) x^2+10...

Factorize the following:
(a) $36 x^{3} y^{2}-8 x^{2} y$
(e) $x^{2}+10 x+24$
(b) $x^{3}+3 x^{2} y+2 x y^{2}+6 y^{3}$
(f) $x^{2}-10 x+16$
(c) $4 x^{2}+12 x+9$
(g) $x^{2}-5 x-36$
(d) $(3 x+4 y)^{2}-(2 x-y)^{2}$
(h) $6 x^{2}+5 x-6$

Key Concepts

Greatest Common Factor (GCF)

This is a basic concept in factoring where you identify the highest common factor shared by all terms in an expression. Factoring out the GCF simplifies the polynomial, making it easier to deal with by reducing the expression to a product of the GCF and a simpler polynomial.

Quadratic Factorization

This concept involves rewriting a quadratic polynomial (typically of the form ax² + bx + c) as a product of two binomials. It often uses methods such as finding two numbers that multiply to ac and add to b, completing the square, or applying the quadratic formula if necessary.

Perfect Square Trinomial

A perfect square trinomial is an expression that takes the form a² + 2ab + b² or a² - 2ab + b², which factors neatly into (a + b)² or (a - b)², respectively. Recognizing this pattern simplifies the factorization process by reducing the quadratic to a squared binomial.

Difference of Squares

This pattern occurs when an expression is the difference between two perfect square terms, following the formula a² - b² = (a + b)(a - b). It is a powerful and straightforward tool for factoring expressions that fit this structure.

Factor by Grouping

Factor by grouping is a technique used primarily when dealing with polynomials that have four or more terms. This method involves grouping terms with common factors and factoring them separately, eventually revealing a common binomial factor, which can then be factored out to simplify the expression.

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