Transcript 3.3 Common Factors of a Polynomial

POLYNOMIALS REVIEW
Introduction to section 3.3
WHAT IS A POLYNOMIAL?
• Collection of algebraic terms used to represent
a statement
first term
third term
second term
fourth term
WHAT IS A POLYNOMIAL?
MONOMIAL:
A one-term expression
WHAT IS A POLYNOMIAL?
BINOMIAL:
A two - term expression
WHAT IS A POLYNOMIAL?
TRINOMIAL:
A three - term expression
DEGREE OF A TERM
•
•
Depends upon the exponents of the variable
A term with only one variable has a degree equal to the
exponent of the variable:
What is the degree of this
polynomial?
DEGREE OF A TERM
•
•
Depends upon the exponents of the variable
A term with mote than one variable has a degree equal to
the sum of the exponent of the variables:
What is the degree of this
polynomial?
DEGREE OF A POLYNOMIAL
•
Depends upon the term with the highest degree
2
4
1
DEGREE of this polynomial is 4!
DEGREE OF A POLYNOMIAL
EXAMPLE:
What is the degree of each term? And what is the
degree of this polynomial?
5
6
5
DEGREE of this polynomial is 6!
ADDITION OF POLYNOMIALS
An expression can be simplified by collecting like
terms and adding their coefficients.
1) 20xy2 + 16xy - 10y2x + 22xy = 10xy2 + 38xy
2) (x2 + 6x + 22) + (3x2 + 10x - 16)
3) 7x2 + 16x - 14
3x2 - 10x + 12
10x2 + 6x - 2
= 4x2 + 16x + 6
SUBTRACTION OF POLYNOMIALS
Collect like terms and add the additive inverse. Or,
distribute the negative sign throughout the polynomial.
1. (12x2 - 16x + 9) - (7x2 + 22x - 17)
= (12x2 - 16x + 9) - 1(7x2 + 22x - 17)
= 12x2 - 16x + 9 - 7x2 - 22x + 17
= 5x2 - 38x + 26
2. (3x2 + 4x - 18) - (8x2 + 15x + 12) + (2x2 + 5)
= 3x2 + 4x - 18 - 8x2 - 15x - 12 + 2x2 + 5
= -3x2 - 11x - 25
3.
4x2 + 10x - 16
-1(8x2 + 4x - 22)
4x2 + 10x - 16
-8x2 - 4x + 22
-4x2 + 6x + 6
MULTIPLYING AND DIVIDING OF POLYNOMIALS
(6x3y2)(4x3y) = 24x6y3
(-10xy4)(3x2y3) = -30x3y7
16x4y5
= -4x3y4
-4xy
For monomial multiplication and division,
apply the exponent rules.
Examples
1) 12xy2 + 6xy - 130y2x + 29xy
= - 118xy2 + 35xy
2) (7x2 + 61x + 49) + (31x2 + 14x - 1)
= 38x2 + 75x + 48
3. (121x2 - 116x + 98) - (73x2 + 122x - 317) = 48x2 - 238x + 415
4. (11z3m2)(4m3z) = 44z4m5
5. (-n3x4y4)(34nx2y3) = -34 n4x8y7
6. 222x3y7
= -37x2y6
-6xy
3.3 Common factors of a polynomial
• We will model and record factoring a polynomial
ALGEBRA TILES
HOW TO USE THEM?
First of all:
Know what the tiles represent!!!
x
x
x²
x
1
x
1
1
1
Big Square (variable squared tile) = any variable squared (x², y², z²…)
Rectangle (variable tile): = any variable (x, y, z, m…)
Small Square (unit tile) = number one
HOW TO USE THEM?
Positive x² tile
Positive x tile
Negative x² tile
Negative x tile
+1
-1
Big Square (variable squared tile) = any variable squared (x², y², z²…)
Rectangle (variable tile): = any variable (x, y, z, m…)
Small Square (unit tile) = number one
HOW TO USE THEM?
By combining the tiles we can easily model
polynomials and their operations
I will be using these tiles:
x²
x
1
x²
x
1
HOW TO USE THEM?
How do you record these tiles in symbols (numbers)?
x²
x²
3x²
x²
HOW TO USE THEM?
How do you record these tiles in symbols (numbers)?
1
-x
-x²
-x² + (-x) + 1 = -x² - x + 1
HOW TO USE THEM?
How do you record these tiles in symbols (numbers)?
4
4x
-x² + 4x + 4
-x²
What is the given polynomial?
YOU have 20 seconds for each example!!
What is the given polynomial?
-2x² + 5x - 5
What is the given polynomial?
4x² -5x - 5
What is the given polynomial?
-2x² + 5x + 4
What is the given polynomial?
-2x² - 3x - 3
What is the given polynomial?
4x² - 5x - 2
What is the given polynomial?
2x² - 1
Represent the Polynomial
NOW, do the opposite! Try to come up with the pictorial forms for a
given polynomial!
3x² - 5x +2
-6x² + 2
-x² - x + 9
4x² + 9x - 9
- 6x + 13
Represent the Polynomial
3x² - 5x +2
Represent the Polynomial
-6x² + 2
Represent the Polynomial
-x² - x + 9
Represent the Polynomial
4x² + 9x - 9
Represent the Polynomial
- 6x + 13
MULTIPLYING A POLYNOMIAL BY A
MONOMIAL
Factor this polynomial ALGEBRAICALLY
8x + 4
1(8x + 4)
2(4x + 2)
4(2x + 1)
MULTIPLYING A POLYNOMIAL BY A
MONOMIAL
Come up with a pictorial form for this
polynomial!
8x + 4
MULTIPLYING A POLYNOMIAL BY A
MONOMIAL
8x + 4
MULTIPLYING A POLYNOMIAL BY A
MONOMIAL
• Sketch or show all the ways you can arrange these tiles to form a
rectangle!
• Beside each sketch, write the multiplication sentence it represents.
MULTIPLYING A POLYNOMIAL BY A
MONOMIAL
POSSIBILITY #1
MULTIPLYING A POLYNOMIAL BY A
MONOMIAL
POSSIBILITY #2
MULTIPLYING A POLYNOMIAL BY A
MONOMIAL
POSSIBILITY #3
MULTIPLYING A POLYNOMIAL BY A
MONOMIAL
• The dimensions of each rectangle represent the factors
of the polynomial.
4x + 2
2(4x + 2)
2
MULTIPLYING A POLYNOMIAL BY A
MONOMIAL
• The dimensions of each rectangle represent the factors
2x + 1
of the polynomial.
4(2x + 1)
4
MULTIPLYING A POLYNOMIAL BY A
MONOMIAL
• The diagrams above show there are 3 ways to factor
the expression 8x + 4:
MULTIPLYING A POLYNOMIAL BY A
MONOMIAL
1(8x + 4)
2(4x + 2)
4(2x + 1)
The first two ways:
are INCOMPLETE because the
second factor in each case
can be factored even further!
MULTIPLYING A POLYNOMIAL BY A
MONOMIAL
1(8x + 4)
2(4x + 2)
4(2x + 1)
The third way:
is COMPLETE because the
second factor in the third case
can’t be factored further!
MULTIPLYING A POLYNOMIAL BY A
MONOMIAL
1(8x + 4)
2(4x + 2)
4(2x + 1)
We say that 8x + 4 is
factored fully when we write
8x + 4 = 4(2x + 1)
Factor a binomial 6c + 4c² completely.
6c + 4c²
Draw all the tiles that you will need
Factor a binomial 6c + 4c² completely.
6c + 4c²
Arrange algebra tile in a rectangle
Factor a binomial 6c + 4c² completely.
2c + 3
2c
Factor a binomial 6c + 4c² completely.
2c + 3
2c
Factor a binomial 6c + 4c² completely.
6c + 4c² = 2c (2c + 3)
POWERPOINT PRACTICE EXAMPLE
Factor binomials 3g + 6 and 8d + 12d² completely.
Factor a binomial 5 – 10z – 5z² completely.
•
•
When a polynomial has negative terms or 3 different
terms, we cannot remove a common factor by
arranging the tiles as a rectangle.
Instead, we can sometimes arrange the tiles into equal
groups.
Draw all the tiles that you will need
5 – 10z – 5z²
5 – 10z – 5z²
Arrange all these tiles into equal groups!
5 – 10z – 5z²
Arrange all these tiles into equal groups!
• There are 5 equal groups and each group contains the
trinomial 1 – 2z - z².
• So, the factors are 5 and 1 – 2z - z².
POWERPOINT PRACTICE EXAMPLE
Factor the trinomial. Verify that the factors are correct.
6 – 12z + 18z2
Factoring Polynomials in More than One Variable
-12x³y – 20xy² - 16x²y²
Factoring Polynomials in More than One Variable
-12x³y – 20xy² - 16x²y²
•
Write the polynomial as a product of the Common
Factor and the remaining factors
4xy(-3x² – 5y – 4xy)
-4xy(3x² + 5y + 4xy)
Factoring Polynomials in More than One Variable
-12x³y – 20xy² - 16x²y²
•
Write the polynomial as a product of the Common
Factor and the remaining factors
4xy(-3x² – 5y - 4xy)
POWERPOINT PRACTICE EXAMPLE
Factor the trinomial. Verify that the factors are correct.
-20c4d - 30c3d2 - 25cd