Home | Math | Advanced Factorization

November 5, 2023
written by Rida Mirza

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

## 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 âˆ’ 3x2 + 4x âˆ’ 12

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

x3 âˆ’ 3x2 + 4x âˆ’ 12 = ( x3âˆ’3x2 ) + ( 4xâˆ’12 )

Here,Â x2Â is common in the firstÂ 22Â terms andÂ 44Â is common in the lastÂ 22Â terms. Factor them out!

( x3âˆ’3x2) + ( 4xâˆ’12 ) = x2 ( xâˆ’3 )+4 ( xâˆ’3 )

Now, factor outÂ (xâˆ’3).

= ( xâˆ’3 )( x2+4 )

## FAQs

### What is the best method to start factoring a difficult polynomial?

The best first step is often applying the rational root theorem to get possible rational roots, and then using the factor theorem to test these roots.

### When should I use synthetic division versus long division?

Synthetic division is quicker for determining if a linear factor divides a polynomial. Use long division if you need to find the actual quotient or remainder.

### Can every polynomial be factored into linear factors?

No, some polynomials of degree 5 or higher are “irreducible” and cannot be factored into linear factors with integer or rational coefficients. In those cases, factorization may require complex roots.

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