Factoring Review...
Factoring involves writing a sum or difference of monomials as a product of polynomials.
We will review factoring techniques learned in other math courses.
three types we will look at today:
­factor by taking out(removing) a common factor
­factoring a difference of squares
­factoring a sum/product polynomial
For each, factor by removing the greatest common factor.
Factor using difference of squares
­both terms must be perfect squares (in more advanced factoring any difference can be
used)
­easy to see...only 2 terms
Sum/Product trinomials
­what are two numbers that add up to 13 and multiply to give 42?
­if you can do this you can factor quadratics that have an a= 1
Guidelines for Factoring a polynomial expression.
If we are asked to factor a polynomial expression, the following guidelines shoudl help us to determine
the best method
1. Look for a common factor. If there is one, take out the common factor and look for further
factoring.
2. If there is a binomial expression, look for a difference of squares.
3. If there is a trinomial expression of the form x2+bx+c, look for sum/product.
4. Is there is a trinomial of the form ax2+bx+c, look for factoring by decomposition (later, but look
for common factor first!)
5. After factoring, check to see if there is further factoring possible.
In flowchart form!!
If binomial,
look for diff. of
squares
a2­b2
Look for a common factor
If trinomial...
ax2+bx+c
a=1
sum/product
a≠1
decomposition
Factoring by decomposition: polynomials of type: ax2+bx +c, a≠1
Expand the following
We expanded (2x+1)(3x+4) to:
6x2+3x+8x+4 or 6x2+11x+4
To factor 6x2+11x+4 , we have to break 11x up into 3x and 8x
How do we know to do this?
Sum and product! (kinda)
Factor:
You try:
Earlier we have looked at factoring quadratics like:
Today we are going to look at quadratics where x is not just a variable, but an
expression, or function of x, f(x).
We use the same methods to factor a polynomial of form:
To start we will restrict f(x) to be a monomial (one term).
In the trinomial x2+bx+c, the degrees of the terms are 2, 1 and zero. We can use the
sum/product method for any multiples of these terms, i.e. 4,2,0; or 6,3,0
You try:
We can use the same idea when it comes to using decomposition...
Factor:
You try:
Factoring trinomials (f(x))2+b(f(x))+c:
We can use the same concept when f(x) is a binomial...
factor:
More complex factoring...
difference of squares (f(x))2­(g(y))2 and a2x2 ­ b2y2
Expand
Factor
(ax­by)(ax+by)
a2x2 ­ b2y2
Factor:
Fractional difference of squares...
To take the square root of a fraction...take the root of the numerator and the root of
the denominator.
Using substitution...Factor:
Where f(x) and g(y) are binomials...factor...
Fractional and "negative a" Factoring
The first thing you must do when you factor any quadratic is look for a common
factor. If a<1 (fraction or negative) you must factor something out first!
Factor:
grab a textbook!!
page 229­230
#1ab, 2ab, 3, 4ac, 5, 6
Factor Theorem: Used to test whether or not a given binomial
is a factor of a given polynomial function
is a factor of
iff
indicates that
Ex: Is
a factor of
of
?
?
Possible factors will be factors of the constant term (c)
is a possible factor of
Using the factor theorem, determine if each of the following linear polynomials is a factor of P(x)
1. Let P(x) = 3x3 ­ 5x2 ­ 6x + 8 .
(iii) x + 3
(ii) x ­1
(i) x +1
2. Let P(x) = 9x3 ­ 21x2 ­ 56x ­16 .
(iii) x ­1
(ii) x +1
(i) x ­ 4
3
3. Let P(x) = x ­ 7x + 6 .
(i) x + 2 (ii) x ­1 (iii) x + 3
4. Let P(x) = 4x3 ­11x2 +10x ­ 3 .
(i) x ­1 (ii) x + 2 (iii) x ­ 3
5. Let P(x) = 3x3 + 5x2 ­ 6x ­ 8 .
(i) x +1 (ii) x ­1 (iii) x + 2