Ch 9 – Polynomials
9.1 – Polynomials
Monomial:
Example: Determine whether each expression is a monomial. Explain why or why not.
a2b3c
1
x
5z 3
x2
Polynomial:
Binomial:
Trinomial:
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
3x2 4x3
5a 9 3
Writing Polynomials:
Degree of a Monomial:
Degree of a Polynomial:
Example: Find the degree of each polynomial.
5a2 3
6x2 4x2 y 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?
9.2 – Adding and Subtracting Polynomials
Adding and Subtracting Polynomials:
Examples: Find each sum.
(3s + 4t) + (6s – 2t) (3x + 9) + (5x + 3)
(b2 + 4b – 6) + (3b2 – 3b + 1) (7m2 – 6) + (5m – 2)
(2d2 + 7de – 8e2) + (-d2 + 8e2)
Subtracting Integers:
Examples: Find each difference.
(6x + 5) – (3x + 1) (2y2 – 3y + 5) – (y2 + 2y + 8)
(2g + 7) – (g + 2) (4p2 – p) – (8 + 3p – p2) (4a2 – 3a + 4) – (a2 + 6a + 1)
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.
9.3 – Multiplying a Polynomial by a Monomial
Multiplying a Polynomial by a Monomial:
Examples: Find each product.
x(x + 1) b(2b2 + 3)
g(3g2 + 4) -y(2y – 6)
y(y + 5) b2(2b2 – 4b – 9)
Example: Solve each equation.
11(y – 3) + 5 = 2(y + 22) 3(d – 4) – 8 = 5(5d + 1) – 3
w(w + 12) = w(w + 14) + 12 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.
9.4 – Multiplying Binomials
Multiplying Binomials:
Example: Find each product.
(x + 3)(x – 4) (a + 2)(2a – 3)
(x – 1)(x + 5) (2y – 1)(y – 3)
FOIL:
Example: Find each product.
(d + 2)(d + 8) (n + 3)(n + 5)
(e + 4)(2e – 4) (a – 2)(a2 – 4)
(5x + y)(4x – 2y) (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.
9.5 – Special Products
Square of a Sum and Square of a Difference:
Example:
(b + 5)2 (2d + 1)2
(c – 3)2 (3e – 3)2
Example: Biologist use a method that is similar to squaring a sum to find the characteristics of offspring based on genetic information.
In a certain population, a parent has 10% chance of passing the gene for brown eyes to its offspring. If an offspring receives one eye-color 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 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 probably that an offspring will receive at least one of these genes?
Product of a Sum and a Difference:
Example:
(3 + a)(3 – a) (x + y)(x – y)
(5b – 2)(5b + 2) (5m – 6n)(5m + 6n)