8.5 Factorising polynomials: $x^2 + bx + c$
Having learned about the basics of the Polynomials, Operations of polynomials, and Prime factorization in previous chapters, we are now more equipped to proceed to a more complex concept. In this chapter we will proceed to learning about factoring polynomial expressions.
Polynomials can be factored by looking at the common factors. So if we are given $15x^4 + 3x^3 + x^2$, using the knowledge we have about factoring we can be able to deduce that the common factor for each term in this expression is $x^2$, thus the remaining part of the equation will be ($15x^2 + 3x + 1$) which can be factored out again.
Factoring can also be done by grouping the polynomials. This would test our understanding of the FOIL method which we use in multiplying one polynomial to another. So if we are given an expression 3x + 7y – 21 – xy, we would get the factors, (3y) and (x7) by grouping (3x xy) and (7y21). We could always counter check the answers by applying the FOIL method.
In the proceeding parts of the chapter, after looking at the different ways to factor polynomial expressions, we will then look at the polynomial expressions: $x^2 + bx +c$, and $ax^2 + bx +c$ (the quadratic equation). We are going to learn how to factor these two expressions and also solve for b and c. There are also a number of applications for these two forms of polynomial equations so we would look into that as well.
We will be slightly talking about special products since Perfect Square Trinomials and Difference of Squares will also be tackled, specifically on how these should be factored. After we learn all of the basics about factoring polynomials, we will be answering exercises to test our understanding.
Factorising polynomials: $x^2 + bx + c$
Lessons

1.
Factor the following

2.
Factor with common factoring first

3.
Factor with unusual exponents