Factoring

1
Factoring

• Given a polynomial, can we find factors which
  when multiplied yield the given polynomial? This
  process is called factoring.
• We want to factor polynomials with integer
  coefficients into a product of polynomials of
  lower degree with integer coefficients.
• We want to factor polynomials with rational
  coefficients into a product of polynomials of
  lower degree with rational coefficients.
• The most basic way to factor a polynomial is to
  spot a common factor in each of its terms. For
  example,

2
Factoring by Grouping

• It is possible to discover common factors by
  first grouping terms. This is a "trial and
  error" approach, and it requires pattern
  recognition and practice.
• Problem. Factor Solution.
  First, group the terms containing a.

3
Factoring Second-Degree Polynomials in x

• Problem. Factor Solution. The
  term 2x2 can only come from the factors 2x and x,
  so we write The constant term,
  6, must be the product of factors of opposite
  signs, so we write
  The integer factors of 6 are
  By trying these factors of 6, we find that

4
Sum and Difference of Two Cubes

• Two more formulas which are useful for factoring
• To use these formulas in factoring, we must
  recognize an expression as the sum (or
  difference) of two cubes.
• Example. Factor First we
  note that Then we have

5
Special Factors

• By using the "Special Products and Factors"
  formulas from the previous section and this
  section of the textbook, we can easily factor
  certain polynomials.
• Problem. Factor Solution. We recognize
  this expression as the "difference of two
  squares" so we can apply
  Thus,
• Problem. Factor
  Solution. We recognize this expression as the
  "sum of two cubes" so we can apply
  Thus,

6
More about Factoring

• The previous techniques can be combined.
  Usually, we first try to find a common factor.
• Problem. Factor Solution.
  Since 2x is a common factor, we can
  write The second factor
  is the "difference of two squares" and we have
• Some polynomials cannot be factored. They are
  called irreducible or prime. For example,
  is irreducible over the integers.

7
Summary of Factoring We discussed

• The general idea of factoring
• Factoring by finding a common factor
• Factoring by grouping
• A technique for factoring second degree
  polynomials over the integers
• Sum and difference of two cubes
• The use of "special products" in factoring
• Irreducible or prime polynomials