Download Document

+ +
+ - - - x x x
Adding, Subtracting,
Multiplying, and Factoring
Polynomials
Click on a type above to review.
(x+4)(3x - 1)
Adding Polynomials
• Add like terms together
• Write in standard form
Subtracting Polynomials
• Change subtraction to add the opposite
• Follow adding rules
Multiplying Polynomials
• Use FOIL when multiplying two binomials.
• Use the Box Method.
Like Terms
Terms with the same variable and same exponent.
Which of the following terms are like terms?
2x and -13x
2x2 and -13x
8y4 and -8x4
x3 and x2
Like Terms
Terms with the same variable and same exponent.
Which of the following terms are like terms?
2x2 and -13x
These are not terms because,
although they have the same
variable, the exponents are
different.
Like Terms
Terms with the same variable and same exponent.
Which of the following terms are like terms?
x3 and x2
These are not terms because,
although they have the same
variable, the exponents are
different.
Like Terms
Terms with the same variable and same exponent.
Which of the following terms are like terms?
8y4 and -8x4
These are not terms because they
do not have the same variable.
Standard Form of a Polynomial
Polynomial is written in standard form when the
terms are in order from greatest exponent to least
exponent.
Example:
5x3 + 10x2 + 6x - 8 is in standard form.
-10x2 + 7x + 4x3 + 3 is not in standard form.
Add the Opposite
• To add the opposite, change subtraction to
addition and add the opposite of the term being
subtracted.
• If you are subtracting a polynomial in
parentheses, change everything the second
parentheses to the opposite sign.
x
2
 

 2x  1 +
  2 x 2+ x -5
FOIL Method
(x+7)(x - 4)
• Multiply the FIRST terms in each binomial
(x+7)(x + -4)
• Multiply the OUTER terms in each binomial
(x+7)(x + -4)
• Multiply the INNER terms in each binomial
(x+7)(x + -4)
• Multiply the LAST terms in each binomial
(x+7)(x + -4)
• Add like terms and write in standard form
Box Method
(x+7)(x - 4)
• Draw a box.
x
7
• Write one factor on one side
x
x2
7x
and write the other factor on
the other side.
-28
-4 -4x
• Find the area of each small
box.
• Add like terms of the small
x2 + 7x + -4x + -28
box areas, and write in
( x+7)(x-4)=x2 + 3x + -28
standard form.
Factoring Polynomials
Click on a form to review how to factor
x2+bx+c form
ax2+bx+c form
Factoring x2 + bx + c form
• Find two factors of c whose sum is b.
• Example:
To factor x2 + 7x + 12,
find factors of 12 whose sum is 7:
(choose one below)
a. 6 and 2
b. -3 and -4
c. 3 and 4
Oops! Nice try!
You found factors of 12, but their
sum is not 7.
Try again!
A binomial is a polynomial with
two terms.
Which of the following are binomials?
12x2
4x + 1
7x2 - 3x + 2
-14x4 - 2
A binomial is a polynomial with
two terms.
Which of the following are binomials?
12x2
This is a monomial
because there is only
one thing being added.
A binomial is a polynomial with
two terms.
Which of the following are binomials?
7x2 - 3x + 2
This is a trinomial
because there are three
things being added.
Great job! 3 and 4 are factors of
12, and they add up to 7.
Factoring
2
x +
Factoring
2
x
bx + c form
+ 7x + 12
Since 3 and 4 are the factors of 12, the
factors of the polynomial are
(x + 4) and (x + 3)
So,
x2 + 7x + 12 = (x + 4)(x + 3)
Factoring ax2+bx+c form
• Multiply the first and last terms.
• Find factors of that term whose sum is bx.
Example:
To factor 2x2+11x+5,
multiply 2x2 and 5 to get 10x2.
Then find factors of 10x2 whose sum is 11x.
(Click on the correct factors below.)
a. 5x and 2x b. 10x and x
c. -10x and -1x
Oops! Nice try!
You found factors of
10x2, but their sum is
not 11x.
Try again!
Great job! 10x and x
are factors of 10x2,
and they add up to 11x.
Go on to the next step.
Factoring 2x2+11x+5
• Now that you’ve found the two factors,10x and
x, make a box.
• Put the first term of the polynomial in the first
square
• Put the last term in the polynomial in the last
square
• Put the two factors in the 2nd and 3rd square.
2x2
10x
x
5
Factoring 2x2+11x+5
• Now factor out the common factor in each
row and column.
x
5
2x
2x2
10x
1
x
5
• The factors are the sides of the box.
2x2+11x+5 = (x + 5)(2x + 1)
Now try one
on your own
Factor 3x2 + 10x + 8
Which of the following are the factors?
a. (3x + 2)(x + 4) b. (3x + 8)(x +1)
c. (3x + 4)(x + 2)
You are ready to move on to the maze!
I love algebra!
Pick a path to begin from.
Path 1
x
2
 
 3x  1  4 x 2  5 x  6

a. 5x2 - 2x + 5 b. 5x2 + 8x + 5 c. 5x2 + 2x + 5
Path 2
3x
a. 5x2+2x+4
Or
2
 

 2x - 1  2x - 4x  5
2
b. 5x2-2x+4
c. 5x2+6x+4
4x
a. x2 - 7x + 6
2
 
 6 x  1   3x 2  x  7
b. x2 + 7x + 6

c. 7x2 - 7x + 8
( x 2  3x  1)  (4 x 2  5x  6)
a. -3x2 - 2x+5
b. -3x2 - 2x + 7
c. -3x2 +8x - 7
3x
a. x2 - 2x - 6
2
 
 2x 1  2x2  4x  5
b. x2 + 6x - 6

c. x2 - 6x + 4
x  72x - 4
a. 3x2 + 10x - 28
b. 2x2 + 10x - 28
c. 3x2 + 18x - 28
(3 x  3)( x - 5)
a. 4x2 + 12x - 15
b. 3x2 + 12x + 15 c. 3x2 - 12x - 15
( x  2)( 5x 2  3x - 4)
a. 5x3 + 3x2 - 4x
b. 5x3 + 10x2 + 6x - 8
c. 5x3 + 13x2 + 2x - 8
( x  3)( 4 x 2 - 2 x - 1)
a. 4x3 - 2x2 - 3
b. 4x3+10x2 -5x-3
c. 4x3+10x2 - 7x -3
5 x 4 (- x 2  2 x - 7)
a. -5x6 +10x4-35x4 b. 5x6+10x5-35x4 c. -5x6+10x5-35x4
Go back and try again!
3x 2 (- x 5 - 4 x 3 - 2)
a. -3x7-12x5-6x2
b. 3x7-12x5-6x2
c. 3x7-x5-6x2
(7m - 2) 2
a. 49m2 - 14m - 4
b. 49m2 - 28m + 4
c. 49m2-14m+4
(4w - 5) 2
a. 16w2-20w-25
b. 16w2 - 40w + 25
c. 16w2 - 40w - 25
Solve ( x - 3)( x - 2)  0
a. 2, 3
b. 3, -2
c. -3, -2
Solve ( x  7)( x - 3)  0
a. 3, -7
b. 7, -3
c. -7, -3
Factor x - 5 x  6
2
a. (x+3)(x+2)
b. (x-3)(x-2)
c. (x-3)(x+2)
Factor x  4 x - 21
2
a. (x - 7)(x + 3)
b. (x - 7)(x - 3)
c. (x + 7)(x - 3)
2
Factor x - 2 x - 35
a. (x-7)(x-5)
b. (x+7)(x-5)
c. (x-7)(x+5)
Factor x  5 x  6
2
a. (x + 6)(x - 1)
b. (x - 3)(x - 2)
c. (x + 3)(x + 2)
Factor 12 x 2  7 x  1
a. (4x+3)(x+1)
b. (4x-1)(3x-1)
c. (4x+1)(3x+1)
OOPS!
Go back and try again!
Solve by factoring 4 x 2  36  0
a. -3, 3
b. 3
c. 9, 4
Solve by factoring 9 x 2  36  0
a. -2, 2
b. -2
c. -6, 6
Solve by factoring 25x 2  100  0
a. 2, -2
b. -2
c. -10, 10
Factor completely 2 x 4  8x 3  8 x 2
a. 2x2(x+2)(x-2)
b. 2x2(x-2)2
c. 2x2(x2 +4x+ 2)
Factor completely 4 x3  12 x 2  8x
a. 4x(x2-3x+2)
b. 4x(x+2)(x-1)
c. 4x(x-2)(x-1)
back and try again!
Factor 5 x  12 x  4
2
a. (5x+4)(x+1)
b. (5x+2)(x+2)
c. (5x+6)(x+2)
( x  3)(4 x 2  2 x  1)
a. 4x3+10x2-7x-3 b. 4x3+10x2+5x-3 c.4x3-10x2+7x+3
Solve ( x  3)( x  7)  0
a. -7, -3
b. -7, 3
c. -3, 7
Factor 15 x  2 x  1
2
a. (5x - 1)(3x + 1)
b. (5x + 1)(3x - 1)
c. (5x - 1)(3x - 1)
Factor 12 x  7 x  1
2
a. (4x - 1)(3x - 1)
b. (4x + 1)(3x + 1)
c. (4x + 3)(x + 1)
Solve the equation x 2  13x  - 40
a. 8, 5
b. 5, -8
c. -5, -8
Solve by factoring 18 x 2  72
a. -2, 2
b. -6, 6
c. -2
Factor x 2  12 x  35
a. (x - 7)(x - 5)
b. (x + 7)(x + 5)
c. (x + 7)(x - 5)
x
a. 5x2 - 2x + 5
2
 
 3x  1  4 x 2  5 x  6
b. 5x2 + 8x + 5

c. 5x2 + 2x + 5
3x
a. 5x2+2x+4
2
 

 2x - 1  2x - 4x  5
b. 5x2-2x+4
2
c. 5x2+6x+4
You’ve made it to the end.
You are so intelligent!
Now go back and try the other path.