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