Advanced Factorization

Factorization is an important topic in algebra. It involves breaking down polynomials into a product of linear factors.

In this article, we will discuss several advanced factorization methods used to factor higher-degree polynomials.

Factor Theorem

The factor theorem is a useful tool for factoring polynomials. It states that if (x – a) is a factor of the polynomial f(x), then f(a) = 0.

Example

Consider the polynomial,

f(x) = x³ – 2x² – 5x + 6

To use the factor theorem, we set f(a) = 0 and solve for a:

f(2) = (2)³ – 2(2)² – 5(2) + 6 = 0

Since f(2) = 0, we know (x – 2) is a factor.

Grouping Method

The grouping method is used to factor quadratic trinomials of the form:

ax² + bx + c

To use grouping, we split the middle term bx into two terms that add to bx. Then we factor by grouping:

Example

x² – 5x + 6

Split -5x into -2x – 3x:

= x² – 2x – 3x + 6

Factor each pair:

= x(x – 2) – 3(x – 2)

Take common factor:

= (x – 2)(x – 3)

So the factors are (x – 2) and (x – 3).

Completing the Square

Completing the square allows us to factor quadratic expressions of the form:

ax² + bx

To complete the square, we add and subtract the square of half of b.

Example

x² + 10x

Half of 10 is 5, and 5^2 is 25. Add and subtract 25:

= x² + 10x + 25 – 25

This forms a perfect square trinomial:

= (x + 5)² – 25

Factored:

= (x + 5)(x + 5)

So the factors are (x + 5) and (x + 5).

Cubic Formula

The cubic formula gives roots to cubic polynomials of the form:

ax³ + bx²+ cx + d

The cubic formula is more complex than the quadratic formula but allows you to directly solve for the roots. These roots are the linear factors of the cubic polynomial.

Example

x³ − 3x² + 4x − 12

Group the first 22 terms and the last 22 terms together.

x³ − 3x² + 4x − 12 = ( x³−3x² ) + ( 4x−12 )

Here, x² is common in the first 22 terms and 44 is common in the last 22 terms. Factor them out!

( x³−3x²) + ( 4x−12 ) = x² ( x−3 )+4 ( x−3 )

Now, factor out (x−3).

= ( x−3 )( x²+4 )

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