Algebra Honors ( Periods 6 & 7)

Monomial Factors of Polynomial 5-3

In the first chapters we reflected that if a, b, and c, were real numbers and c was not equal to 0,
then
(a + b)/c = a/c + b/c
It also applies to monomials
(5m + 35)/ 5 = 5m/5 + 35/5... which simplifies to  m + 7

To divide a polynomial by a monomial, divide each term of the polynomial by the monomial and then add the results.
From this point on, our textbook has us assume that not divisor equals 0

Divide the following
26uv - 39v
    13v

You can separate each part so that you have
(26uv)/13v  - 39v/13v
which simplifies to
2n -3

Divide

3x⁴ - 9x³y + 6x²y² 
-3x²

This time you notice that -3x² is a factor of all three terms of this polynomial
and you can separate the polynomial into three separate terms or... as taught in class you can easily do each ( carefully)
so that you simplify the fraction to
-x² + 3xy -2y²

Divide:
x³y - 4y + 6x
     xy

Here you might want to show the three terms to see what is happening to each individual term...

x³y 
  xy

-4y
xy

+6x
xy

which simplifies to

x² -4/y + 6/x

You definitely could simplify by crossing out the common factors  each term shares with the divisor.

One polynomial is evenly divisible or just divisible by another polynomial if the quotient is also a polynomial.
So the first two examples show divisibility but this last one does NOT.

You factor a polynomial by expressing it as a product of other polynomials. The factor set for a polynomial have integral coefficients.
You can use division to test for factors!

The greatest monomial factor of a polynomial is the GCF of its terms!

Factor
5x² + 10x
The greatest monomial factor is 5x
You don't want to change the value of your polynomial-- you just want to factor it!
Pull out the 5x and divide by 5x as well.. because you are NOT CHANGING the VALUE
Its simply using the Distributive Property

5x(5x² + 10x)
     5x

5x(x + 2)
To check-- just multiply out using your knowledge of the distributive property!!

Factor
4x - 6x³ + 14x

The greatest monomial factor is 2x

2x(4x - 6x³ + 14x)
   2x
2x(2x - 3x² + 7)

Factor
8a²bc² - 12ab²c²
The greatest monomial factor is 4abc²


4abc²(8a²bc² - 12ab²c²)
  4abc²

4abc²(2a-3b)

Practice these and you will be able too do the division steps mentally.
Check your factorization by multiplying the resulting factors!
Make sure you end up with where you started-- when you check!!