Download Polynomial Packet Notes - Magoffin County Schools

Unit Essential Questions
Are two algebraic expressions that appear to be different actually
equivalent?
What is the relationship between properties of real numbers and
properties of polynomials?
MACC.912.A-APR.A.1: Understand that polynomials
form a system analogous to the integers, namely, they
are closed under the operations of addition,
subtraction, and multiplication; add, subtract, and
multiply polynomials.
MACC.912.A-SSE.A.1a: Interpret parts of an
expression, such as terms, factors, and coefficients.
WARM UP
Simplify.
1) (3x + 2y) + 5x
8x + 2y
2) (4h + 5 j) - 3h
h + 5j
3) (-6a + 5b) + 2a
–4a + 5b
KEY CONCEPTS AND
VOCABULARY
A monomial is a real number, a variable, or the product
of real numbers and variables (Note: the variables must
have positive integer exponents to be a monomial).
The degree of a monomial is the sum of the exponents
of its variables.
A polynomial is a monomial or a sum of monomials.
Standard form of a polynomial means that the degrees
of its monomial terms are written in descending order.
The degree of a polynomial is the same as the degree of
the monomial with the greatest exponent.
KEY CONCEPTS AND
VOCABULARY
0
1
2
3
CLASSIFICATION OF POLYNOMIALS
DEGREE
NUMBER OF TERMS
Constant
1
Monomial
Linear
2
Binomial
Quadratic
3
Trinomial
Cubic
4
Polynomial with 4 terms
EXAMPLES OF
MONOMIALS
EXAMPLES OF NOT
MONOMIALS
6
x -3
p/7
4/y
4.75a2bc 3
g
9s 2t -5
8xy / z
EXAMPLE 1:
IDENTIFYING POLYNOMIALS
Determine whether each expression is a polynomial. If
it is a polynomial, classify the polynomial by the degree
and number of terms.
2
3
a) 2x - 3x + 4x
b) 12x 2 +10x -3
Polynomial,
Cubic
Trinomial
c) x 2 + 3x + 4x 2
Polynomial,
Quadratic
Binomial
Not a Polynomial
d) 5
Polynomial,
Constant
Monomial
EXAMPLE 2:
WRITING POLYNOMIALS IN STANDARD FORM
Write the polynomial in standard form. Then identify
the leading coefficient.
a) 4a2 - 7a + 3a5
3a5 + 4a2 - 7a
Leading Coefficient = 3
b) 5h - 9 - 2h 4 - 6h 3
-2h 4 - 6h 3 + 5h - 9
Leading Coefficient = –2
c) -2 + 6t 2 - 7t + 2t 2
8t 2 - 7t - 2
Leading Coefficient = 8
EXAMPLE 3:
ADDING POLYNOMIALS
Simplify.
2
2
a) (2x - 7 + 5x) + (-4x
+ 6x + 3)
-2x 2 +11x - 4
b)
(5x + 7x 2 + 3) + (-5x 2 + x 3 - 4)
x 3 + 2x 2 + 5x -1
EXAMPLE 4:
SUBTRACTING POLYNOMIALS
Simplify.
a) (3x + 2 - x 2 ) - (4x - 5 + 2x 2 )
-3x 2 - x + 7
b) (12x 2 - 8x +11) - (-14 +10x 2 - 6x)
2x 2 - 2x + 25
EXAMPLE 5:
SIMPLIFYING USING GEOMETRIC FORMULAS
Express the perimeter as a polynomial.
a)
b)
3x 2 + 2x - 3
3x 2 - 4x +10
x2 - 4
x 2 - 7x + 2
5x 2
8x 2 -10x - 2
9x - 4x + 6
2
EXAMPLE 6:
ADDING AND SUBTRACTING POLYNOMIALS IN
REAL-WORLD APPLICATIONS
The equations H = 3m +120 and C = 4m + 84 represent the
number of Miami Heat hats, H, and the number of
Cleveland Cavalier hats, C, sold in m months at a sports
store.
a) Write an equation for the total, T, of Heat and
Cavalier hats sold.
T = 7m + 204
b) Predict the number of Heat and Cavalier hats sold in
9 months.
267 Hats
RATE YOUR UNDERSTANDING
ADDING AND SUBTRACTING
POLYNOMIALS
MACC.912.A-APR.A.1: Understand that polynomials form a system analogous to the integers,
namely, they are closed under the operations of addition, subtraction, and multiplication; add,
subtract, and multiply polynomials.
MACC.912.A-SSE.A.1a: Interpret parts of an expression, such as terms, factors, and coefficients.
RATING
4
TARGET
3
2
1
LEARNING SCALE
I am able to
• add and subtract polynomials in real-world applications or in more
challenging problems that I have never previously attempted
I am able to
• identify a polynomial and write polynomials in standard form
• add and subtract polynomials
I am able to
• identify a polynomial and write polynomials in standard form with
help
• add and subtract polynomials with help
I am able to
• identify the degree of a monomial
MACC.912.A-APR.A.1: Understand that polynomials
form a system analogous to the integers, namely, they
are closed under the operations of addition,
subtraction, and multiplication; add, subtract, and
multiply polynomials.
WARM UP
Simplify.
1) x 4 × x 5
x9
2) (3x yz)(-2xy z )
2
-6x 3 y 3z 4
2 3
3 2 3
5x
y z
3)
xyz
5x 2 yz 2
KEY CONCEPTS AND
VOCABULARY
You can use the Distribution Property to multiply a
monomial by a polynomial.
EXAMPLE 1:
MULTIPLYING A POLYNOMIAL BY A MONOMIAL
Simplify.
a) 2x 2 (6x 2 - 2x + 5)
b) -3x 2 (x 2 + 3x - 8)
12x 4 - 4x 3 +10x 2
-3x 4 - 9x 3 + 24x 2
EXAMPLE 2:
SIMPLIFYING EXPRESSIONS WITH A PRODUCT
OF A POLYNOMIAL AND A MONOMIAL
Simplify.
a) 2x 2 (-2x 2 + 5x) - 5(x 2 +10)
-4x 4 +10x 3 - 5x 2 - 50
b) 3(5x 2 + x - 4) - x(4x 2 + 2x - 3)
-4x 3 +13x 2 + 6x -12
EXAMPLE 3:
SIMPLIFYING USING GEOMETRIC FORMULAS
Express the area as a polynomial.
a)
b)
7x
4x
x 2 + 7x -12
x2 + 3
2x 3 + 6x
7x 3 + 49x 2 - 84x
EXAMPLE 4:
SOLVING EQUATIONS WITH POLYNOMIALS ON EACH
SIDE
Solve.
a) 2x(x + 4) + 7 = (x + 9) + x(2x +1) +12
7
x=
3
b) x(x 2 + 3x + 5) + 2x 3 = 3x(x 2 + x + 5) +10
x = –1
RATE YOUR UNDERSTANDING
MULTIPLYING A POLYNOMIAL BY A
MONOMIAL
MACC.912.A-APR.A.1: Understand that polynomials form a system analogous to the integers,
namely, they are closed under the operations of addition, subtraction, and multiplication; add,
subtract, and multiply polynomials.
RATING
4
TARGET
3
2
1
LEARNING SCALE
I am able to
• multiply a polynomial by a monomial in more challenging
problems that I have never previously attempted (such as
solving equations)
I am able to
• multiply a polynomial by a monomial
I am able to
• multiply a polynomial by a monomial with help
I am able to
• understand that the distributive property can be applied to
polynomials
MACC.912.A-APR.A.1: Understand that polynomials
form a system analogous to the integers, namely, they
are closed under the operations of addition,
subtraction, and multiplication; add, subtract, and
multiply polynomials.
WARM UP
Simplify.
1) 9(x - 4)
9x - 36
2) -3(c + 6)
-3c -18
3) 1 (8y +16)
4
2y + 4
KEY CONCEPTS AND
VOCABULARY
METHODS FOR MULTIPLYING POLYNOMIALS
DISTRIBUTIVE PROPERTY METHOD
Example:
(x + 4)(x - 3)
(x + 4)(x - 3)
x(x - 3) + 4(x - 3)
x 2 - 3x + 4 x - 12
x 2 + x - 12
FOIL METHOD
Example:
(x + 4)(x - 3)
EXAMPLE 1:
FINDING THE PRODUCT OF TWO BINOMIALS
USING THE DISTRIBUTIVE PROPERTY
Simplify using the distributive property.
a) (x - 2)(x + 7)
x 2 + 5x -14
b) (2a + 7)(3a - 5)
6a2 +11a - 35
c) (r + 5)(5r +10)
5r 2 + 35r + 50
EXAMPLE 2:
FINDING THE PRODUCT OF TWO BINOMIALS
USING THE FOIL METHOD
Simplify using the FOIL method.
a) (x + 3)(x + 9)
x 2 +12x + 27
b)
(5w - 2)(w + 3)
5w2 +13w - 6
c)
(4k -1)(3k - 7)
12k 2 - 31k + 7
EXAMPLE 3:
FINDING THE PRODUCT OF A BINOMIAL AND
TRINOMIAL
Simplify using the distributive property.
2
a) (2x - 6)(3x + x -1)
6x 3 -16x 2 - 8x + 6
b) (m -1)(m 3 - 4m +12)
m4 - m3 - 4m2 +16m -12
c) (b2 - 4b + 3)(b - 2)
b 3 - 6b2 +11b - 6
EXAMPLE 4:
SIMPLIFYING PRODUCTS
Simplify.
a) (x + 2)[(x 2 + 3x - 6) + (x 2 - 2x + 4)]
2x 3 + 5x 2 - 4
b)
[(x + 3x - 7) - (x - 2x + 6)](x - 4)
2
5x 2 - 33x + 52
2
RATE YOUR UNDERSTANDING
MULTIPLYING POLYNOMIALS
MACC.912.A-APR.A.1: Understand that polynomials form a system analogous to the integers,
namely, they are closed under the operations of addition, subtraction, and multiplication; add,
subtract, and multiply polynomials.
RATING
4
TARGET
3
2
1
LEARNING SCALE
I am able to
• multiply two binomials or a binomial by a trinomial in more
challenging problems that I have never previously attempted
I am able to
• multiply two binomials or a binomial by a trinomial
I am able to
• multiply two binomials or a binomial by a trinomial with
help
I am able to
• understand the distributive property
MACC.912.A-APR.A.1: Understand that polynomials
form a system analogous to the integers, namely, they
are closed under the operations of addition,
subtraction, and multiplication; add, subtract, and
multiply polynomials.
WARM UP
Simplify.
1) (z + 4)(z + 4)
z 2 + 8z +16
2) (y - 3)(y - 3)
y2 - 6y + 9
3) (q + 6)(q - 6)
q 2 - 36
KEY CONCEPTS AND
VOCABULARY
MULTIPLYING SPECIAL CASES
THE SQUARE OF A BINOMIAL
THE PRODUCT OF A SUM AND
DIFFERENCE
(a + b)2 = (a + b)(a + b) = a2 + 2ab + b2
(a + b)(a – b) = a2 – b2
Or
(a – b)2 = (a – b)(a – b) = a2 – 2ab + b2
EXAMPLE 1:
SIMPLIFYING THE SQUARE OF A BINOMIAL
(SUM)
Simplify.
2
a) (x + 2)
x 2 + 4x + 4
b)
(5x + 2)
2
25x 2 + 20x + 4
c)
(x + 5)
2
2
x 4 +10x 2 + 25
EXAMPLE 2:
SIMPLIFYING THE SQUARE OF A BINOMIAL
(DIFFERENCE)
Simplify.
a) (x - 7)2
x 2 -14x + 49
b) (2x -1)2
4x 2 - 4x +1
c) (x 2 - 3)2
x 4 - 6x 2 + 9
EXAMPLE 3:
SIMPLIFYING THE PRODUCT OF A SUM AND
DIFFERENCE
Simplify.
a) (x + 3)(x - 3)
b) (2x + 5)(2x - 5)
x2 - 9
4x 2 - 25
c) (x - 4)(4 + x)
d) (x 2 + 6)(x 2 - 6)
x 2 -16
x 4 - 36
EXAMPLE 4:
SIMPLIFYING MORE CHALLENGING PROBLEMS WITH
SPECIAL CASES
Simplify.
a) (x + 2y)2
x 2 + 4xy + 4y2
æ1
ö
c) çè x + 2 ÷ø
4
b) (a - 6b)(a + 6b)
a2 - 36b2
2
1 2
x +x+4
16
d) (x + 2)(x - 5)(x - 2)(x + 5)
x 4 - 29x 2 +100
EXAMPLE 4:
SIMPLIFYING MORE CHALLENGING PROBLEMS WITH
SPECIAL CASES
Simplify.
e) (2x + 3)(2x - 3)(x +1)
4x 3 + 4x 2 - 9x - 9
f) [(4x +1)(4x -1)] + [(x + 5)2 ]
17x 2 +10x + 24
RATE YOUR UNDERSTANDING
SPECIAL PRODUCTS
MACC.912.A-APR.A.1: Understand that polynomials form a system analogous to the integers,
namely, they are closed under the operations of addition, subtraction, and multiplication; add,
subtract, and multiply polynomials.
RATING
4
TARGET
3
2
1
LEARNING SCALE
I am able to
• simplify special products in more challenging problems that
I have never previously attempted
I am able to
• find the square of a binomial
• find the product of a sum and difference
I am able to
• find the square of a binomial with help
• find the product of a sum and difference with help
I am able to
• understand that there are special rules to simplify the square
of a binomial and the product of a sum and difference