Visualizing Algebraic Patterns(A)

< 28.7 Visualizing Algebraic Patterns | Topic Index | 28.9 Simple Equations >

The idea of visualizing algebraic expressions is a powerful idea. With a slightly different way of visualizing, the idea of factorizing 2nddegree algebraic expressions acquires a different meaning.

We visualized composite numbers as those which have factors. For example, 12 can be written as 3 X 4. We can also say that we can factorize 12 as 3 X 4.

The same idea can be extended to algebraic expressions if we can visualize each term of a second degree algebraic expression as a rectangle.

For example let us take an expression+ 4x + 3.

Let us imagine as a square of side x, 4x as 4 rectangles with sides 1 &amp; x and 3 as 3 squares of side 1X1. Can we arrange these areas into a large rectangle without any gaps or overlaps? It turns out that this is possible. We can make a larger rectangle whose 2 sides measure x+3 &amp; x + 1!

We have factorized x^2 + 4x + 3 as (x+3) (x+1).

We have seen factorization as a puzzle and the equivalent of finding facts of a composite number!

Expressions Which Cannot be Factored

Working visually will also provide a clue to whether an algebraic expression is factorizable! Expressions which cannot be factorized can be thought of as &ldquo;prime&rdquo; expressions.

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