To factor quadratic polynomials of the form \(f(x)=x^2+bx+c\) requires finding two values \(\alpha\) and \(\beta\) such that \(x^2 + bx + c = (x+\alpha)(x+\beta)\). Expanding the right hand side as shown

\(x^2 + bx + c = (x+\alpha)(x+\beta) = x^2 – (\alpha+\beta)x + \alpha\beta \)

By equating both sides of the polynomial reveals that \(b=\alpha+\beta\) and \(c=\alpha\beta\). In other words, \(\alpha,\beta\) are factors of the constant term \(c\) which sum to \(b\). 

Factoring quadratic polynomials of the form \(f(x)=ax^2+bx+c\) can be found considering \(\displaystyle f(x)=a\left( x^2+\frac{b}{a}x+\frac{c}{a}\right)\) and applying the results  mentioned above to the polynomial \(\displaystyle g(x)=x^2+\frac{b}{a}x+\frac{c}{a}\).

Cubic polynomials can be factorized by utilizing the fact that complex roots come in conjugate pairs. In other words, if \(\lambda=a+b\mathbb{i}\) is a root then so is its conjugate \(\bar{\lambda}=a-b\mathbb{i}\). A necessary consequence is that the remaining root \(c\) for the cubic \(f(x)\) must be real number with \((x-c)\) as a factor and a quadratic polynomial \(g(x)\). 

\(f(x)=(x-c)g(x)\)

Once the root \(c\) has been found, then apply the factoring protocols to the quadratic polynomial \(g(x)\) to completely factor the cubic polynomial. 

The Rational Root Theorem is a procedure for determining possible rational roots, from which we can then test to determine roots and factors for a given polynomial. One such procedure for finding these roots is use synthetic division. 
 

Rational Roots Theorem
Let \(f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_2x^2+a_1x+a_0\) be polynomial with integer coefficients. 

If the rational number is \(\displaystyle x=\frac{p}{q}\) is a root for the polynomial \(f(x)\) then 

• • • The integer \(p\) must divides \(a_0\)
    • The integer \(q\) must divide \(a_n\)

Does the given polynomial have a special form

1. Sum of Squares
2. Difference of Squares
3. Sum of Cubes
4. Difference of Cubes

Is it possible to rearrange the terms to find these forms?

Can the terms be strategically grouped to uncover a factorable pattern?