Introduction to Polynomial Factorizations and Equations

1
Introduction to Polynomial Factorizations and
Equations
2
Equations/FunctionsZeros and Roots

• Whenever two polynomials are set equal to each
  other, the result is a polynomial equation.
• Solve the equation below graphically
• We can rewrite the equation so that one side is 0
  and then find the x-intercepts of the graph.
• You must know how to use both the INTERSECT and
  ZERO features of your calculator. Please do NOT
  just use TRACE.

3
Zeros and Roots
4
The Principle of Zero Products

• Informally stated When we multiply two or more
  numbers and the product is 0, one or more of the
  factors must be 0.
• Formally stated
• If ab 0, then a 0, b 0, or both a and b are
  0.
• If a 0 or b 0, then ab 0.

5
Examples

• Solve (x 2)(x 4) 0.
• Solve 3(2x 5) 0.
• Solve 4x(3x 1)(x 2) 0.
• Given that f(a) a(3a 2), find all the values
  of a for which f(a) 0.

6
Factoring Greatest Common Factor

• To factor a polynomial means to write the
  indicated sum and/or difference of terms as a
  product of factors. To factor means to write as
  a product.
• First, identify the greatest common factor of the
  terms of the polynomial (if it exists).
• Rewrite each term of the polynomial as the
  product of the GCF and another factor.
• Use the distributive property to express the
  polynomial as the product of the GCF and a
  polynomial.

7
Common Factors cont.

• Simplifying the original expression and the
  factored expression should produce equal
  expressions.
• Remember, you are writing an indicated sum or
  difference of TERMS as a PRODUCT of factors. Your
  result should contain NO ungrouped addition or
  subtraction signs.

8
Factoring out the GCF
9
Factoring out a Binomial Factor
10
Factoring by Grouping

• In order to factor a four term polynomial, we
  identify a common binomial factor by regrouping
  the polynomial into two groups of two terms each.

11
Factoring by Grouping Method

• a. Factor out the greatest common monomial factor
  from all four terms, if one exists.
• b. Group the terms in the remaining polynomial
  factor so that each group has a common factor. If
  there is no common factor, a different
  arrangement of the polynomial terms needs to be
  examined.
• c. Factor out the GCF in each group.
• d. There should now be a common polynomial
  factor. Factor it out.
• e. Your result should contain no ungrouped
  addition or subtraction signs.

12
Be Careful -