Algebra Honors ( Period 6 & 7)

Multiplying Polynomials by Monomials 4-5
This is just the distributive property
x(x + 3) = x² + 3x

-2x(4x² - 3x + 5)

-8x³ + 6 x² -10x
The book shows you how to multiply using a vertical method but I think using the original method taught with the distributive property works just as well-- if not better.

n(2-5n) + 5(n² -2 ) = 0
2n - 5n² + 5n² - 10 = 0
2n - 10 = 0
2n = 10
n = 5
and in set notation {5}

1/2(6xc + 4) -2(c + 5/2) = 2/3 (9-3c)
3c + 2 - 2c - 5 = 6 - 2c
3c -3 = 6
3c = 9
c = 3
and in set notation {3}

Multiplying Polynomials  4-6
This is just double distributive property or triple distributive property so you really need to understand the DP

You have learned how to use the DP to multiply
2x(3x + 2)
but... what happens if you had instead
(2x +5)(3x + 2)
There are a number of different strategies to simplify this multiplication
(2x +5)(3x) + (2x + 5)(2)

6x² + 15x + 4x + 10

6x² + 19x + 10

You could also use Fireworks—as show in class or FOIL

Fà First
OàOuter
IàInner
Là Last

 (2x + 5)(3x + 2)

The F is the first terms   (2x)(3x)

The O is the Outer terms (2x)(2)

The I is the Inner terms (5)(3x)

the L is the Last Terms (5)(2)

or 6x² + 4x + 15x + 10

6x² + 19x + 10

Example:

(3x-2)(2x²- 5x-4)

The book shows you how to multiply in vertical form, similar to how you multiply multi-digit  numbers.

Read Page 161 if you are interested in reviewing that strategy

Step 1   2x² – 5x – 4                         Step 2                       2x² – 5x – 4                   

3x – 2                                                                         3x – 2             

6x³-15x² – 12x                                                    6x³-15x² – 12x

                                                                                                 -4x² +10x + 8     

Step 3        Add:

  2x² – 5x – 4                                                                        

   3x – 2             6x³-15x² – 12x                                                    

       -4x² +10x   + 8     

6x³ – 19x² -2x + 8

I showed Fireworks and Double Distributive Property with this example as well

but also showed my favorite… The BOX Method

Create a box as big as the polynomials

In this case it’s a 2 by 3

We talked about the order of the polynomials.

Make sure to place them in descending order.

The book terms it decreasing degree of x:

We discussed

x³ -3x² + xy² + 2y³

To see the advantage of rearranging terms, multiply the polynomial

(y +2x)(x³ – 2y³ + 3xy² + x²y)

We then rearranged both polynomials into decreasing degree in terms of x

(2x + y)(x³ + x²y+ 3xy² – 2y³)

Using the BOX Method

we could find the simplified form to be

2x⁴ + 3x³y + 7x²y² –xy³ -2y⁴