Download Polynomials

Ch 9
Polynomials
9.1
Polynomials
Monomial
• A number, variable, or a product of numbers
and variables that have only positive
exponents
• Cannot have an exponent that is a variable
Example
• Determine whether each expression is a
monomial. Explain why or why not.
– a2 b3 c
1
– x.
– 5z-3
– x2
Polynomial
• A monomial or the sum of one or more
monomials
– Each monomial is a term
– Sum: subtraction is just adding the opposite
x 3  x 2  3x  2
Special Polynomials
• Binomial –
– A polynomial with two terms
• Trinomial –
– A polynomial with three terms
Example
• State whether each expression is a polynomial. If
it is a polynomial, identify it as a monomial,
binomial, or trinomial.
– -4x + 2
– 5 + 3x2 + x + 2
– 3x-2 + 4x3
– 5a – 9 + 3
Polynomials
• Polynomial terms are arranged in either
ascending or descending order
– We will arrange our in descending order!!
Degree of a Polynomial
• Degree of a monomial –
– Sum of the exponents of the variables
• Degree of a polynomial –
– Greatest of the degrees of its terms
Example
• Find the degree of each polynomial.
– 5a2 + 3
– 6x2 – 4x2y – 3xy
– 8b4 + 92
– 2ab + 3a2b + 5a4b2
Example
• The expression 14x3 – 17x2 – 16x + 34 can be
used to estimate the number of eggs that a
certain type of female moth can produce. In
the expression, x represents the width of the
abdomen in millimeters. About how many
eggs would you expect this type of moth to
produce if her abdomen measures 3
millimeters?
Example
• The same female moth… how many eggs
would you expect her to lay if her abdomen
measures 2 millimeters?
Assignment
• 1st Assignment:
– P385: 1 – 3, 5 -15
• 2nd Assignment:
– P385: 16 – 69
9.2
Adding and Subtracting Polynomials
Adding and Subtracting Polynomials
• To add or subtract polynomials, combine all
like terms
– Remember: like terms have the exact same
variables
• Including exponents!
Example
• Find each sum.
– (3s + 4t) + (6s – 2t)
– (b2 + 4b – 6) + (3b2 – 3b + 1)
– (2d2 + 7de – 8e2) + (-d2 + 8e2)
– (3x + 9) + (5x + 3)
– (7m2 – 6) + (5m – 2)
Subtracting Integers
• Subtracting is just adding its additive inverse
or opposite
Example
• Find each difference.
– (6x + 5) – (3x + 1)
– (2g + 7) – (g + 2)
– (4a2 – 3a + 4) – (a2 + 6a + 1)
– (2y2 – 3y + 5) – (y2 + 2y +8)
– (4p2 – p) – (8 + 3p – p2)
Example
• The measure of the perimeter of a triangle is
9a + 2b. Two of the sides have lengths of 3a
+ b and 5a. Find the measure of the third
side of the triangle.
Example
• The perimeter of triangle ABC is 7x + 2y. Find
the measure of the third side of the triangle.
Assignment
• 1st Assignment:
– P392: 3 - 17
• 2nd Assignment:
– P392: 18 – 42, 45 – 51
– 9.1/9.2 Wkst
9.3
Multiplying a Polynomial by a
Monomial
Multiplying a Polynomial by a
Monomial
Example
• Find each product.
– x(x + 1)
– g(3g2 + 4)
– y(y + 5)
– b(2b2 + 3)
– -y(2y – 6)
– b2(2b2 – 4b – 9)
Example
• Solve each equation.
– 11(y – 3) + 5 = 2(y + 22)
– w(w + 12) = w(w + 14) + 12
Example
• Solve each equation.
– 3(d – 4) – 8 = 5(5d + 1) – 3
– a(3 + a) – 2 = a(a – 1) + 6
Example
• Find the area of the shaded region in simplest
form.
Example
• Find the area of the shaded region in simplest
form.
Assignment
• 1st Assignment:
– P396: 1, 3 – 16
• 2nd Assignment:
– P396: 18 – 61
9.4
Multiplying Binomials
Multiplying Binomials
• Use distributive property twice!!
• (x + 3)(x – 4)
Example
• Find each product.
– (x – 1)(x + 5)
– (a + 2)(2a – 3)
– (2y – 1)(y – 3)
FOIL
Example
• Find each product
– (d + 2)(d + 8)
o(5x + y)(4x – 2y)
o(n + 3)(n + 5)
– (e + 4)(2e – 4)
Example
• Find the product.
– (a – 2)(a2 – 4)
– (x2 – 4)(x + 3)
Example
• The volume V of a rectangular prism is equal
to the area of the base B times the height h.
Express the volume of the prism as a
polynomial. Use V = Bh.
Example
• Find the volume of a rectangular prism with
base dimensions x and x + 4 units and height x
+ 2 units.
Assignment
• 1st Assignment:
– P402: 4 – 20
• 2nd Assignment:
– P402: 22-50 even, 51-65
9.5
Special Products
Square of a Sum and Square of a
Difference
Example
• (b + 5)2
• (c – 3)2
• (2d + 1)2
• (3e – 3)2
Real-Life Use
• Biologist use a method that is similar to
squaring a sum to find the characteristics of
offspring based on genetic information.
Example
• In a certain population, a parent has a 10%
chance of passing the gene for brown eyes to
its offspring. If an offspring receives one eyecolor gene from its mother and one from its
father, what is the probability that an offspring
will receive at least one gene for brown eyes?
Example
• In a certain population, a parent has a 20%
chance of passing on a certain gene to its
offspring. If an offspring receives one gene
each from its mother and father, what is the
probability that an offspring will receive at
least one of these genes?
Product of a Sum and a Difference
Example
• Find each product.
– (3 + a)(3 – a)
– (5b – 2)(5b + 2)
– (x + y)(x – y)
– (5m – 6n)(5m + 6n)
Assignment
• 1st Assignment:
– P408: 1 – 9
• 2nd Assignment:
– P408: 10 – 33, 35, 37 – 43
Review
• P412: 1 – 59
• P415: 4 – 24