كتاب الطالب Integrated III الرياضيات للصف العاشر منهج انجليزي Reveal الفصل الثالث 2021 - 2022
The Venn diagram shows the set of complex numbers. Notice that all of the real numbers are part of the set of complex numbers
Complex Numbers (a + bi)
Two complex are equal if and only if their real parts are equal and their parts are equal. The Commutative and Assxiaüve Properties of Multiplication and Addition and the Distributive hold true for complex numbers. To add or subtract cornplex combine like terms. That is, combine the real parts. and combine the imaginary parts
Two complex numbers of the form a + bi and a — bi are called complex conjugates The product of complex conjugates is always a real number
A radical expression is in simplest form if no radicands contain fractions and no radicals in the denominator of a fraction. Similarly, a complex number is in simplest form if no imaginary numbers appear in the denominator of a frætion. You can use complex conjugates to simplify a fraction with a complex in the denominator. This process is called rationalizing the denominator
Example 4 Equate Complex Numbers
Use equations relating the real and imaginary parts to solve for x and y
Apply Example 3 Solve an Equation by Factoring
ACCELERATION The equation represents the displacement d of a car traveling at an initial v where the acceleration a is constant over a given time t. find how long it takes a car to accelerate 30 mph to 4S mph if the car moved 60S feet and accelerated slowly at a rate of 2 feet second squared
I What is the task
Describe the task in your own words. Then list any questions that you may have. How can you find answers to your questions
Sample answer: Solve the equation to find the time for the car to accelerate. The acceleration is given in feet second squared and the velocity is given in miles hour. How do I address the difference in units
How will you approach the task ? What have you learned that you can use to help you complete the task ?
Sample answer: Convert the velocity to feet per second. Then substitute the distance, velocity, and acceleration into the formula and solve for time
3 What is your solution
Use your strategy to solve the problem What is the velocity in feet per second ? 44 fps How long it takes the car to accelerate from 30 mph to 45 mph? 11 s
4 How can you know that your solution is reasonable ?
Write About It! Write an argument that can used to defend your solution
Sample answer: The solutions of the are —SS and 11
Because time cannot be negative, t = 11 is the only viable solution in the context of the situation
Online You can cornplete an Extra Example online
Learn Adding and Subtracting Polynomials
A polynomial is a monomial or the sum of two or more monomials. A binomial is the sum of two monomials, and a trinomial is the sum of three monomials. The degree of a polynomial is the greatest degree of any term in the polynomial
Polynomials can be added or subtracted by performing the operations indicated and combining like terms. You can subtract a polynomial by adding its additive inverse
The sum or difference of polynomials will have the same variables and exponents as the original polynomials, but different coefficients Thus, the sum or difference of two polynomials is also a polynomial
A set is closed if and only if an on any two elements of the set produces another element of the same set Because adding or subtracting polynomials resulG in a the set of is closed under the of addition and subtraction
Example 1 Identify Polynomials
Determine whether each expression is a polynomial. If it is a polynomial, state degree of the polynomial
This expression is not a polynomial is not a monomial
This expression is a polynomial because each term is a monomial
Apply Example 7 Write and Simplify a Polynomial Expression
Byron is baking a three tier cake for a birthday party. Each tier will have I the volume of the previous tier. The dimensions of the first tier are shown. Find the total volume of the cake
I What is the task
Describe the task in your own words. Then list any questions that you may have. How can you find answers to your questions
Sample answer. I need to find the total volume of the cake, which is the sum of all 3 tiers. How can I represent the volume of each tier as a polynomial ? Which properties will I need to know ? I can find the answers to my questions by referencing other examples in the lesson
How will you approach the task ? What have you learned that you can use to help you complete the task
Sample answer: I will find and simplify the volume of each tier and then add them I will use the Distributive Property and FOIL method to complete the task
3 What is your solution
use your strategy to solve the problem
What is the volume of each tier
What is the total volume of the cake
How can yml know that your solution is reasonable
Write It! Write an argument that can used to defend your solution
Sample answer: Because all the expressions are based on the expression for the volume of Tier 1, I can check that the expression for Tier 1 is correct. I can factor the expression for volume of Tier 1 to ensure that the factors are the same as the given dimensions
Study Tip
Zeros The real zeros occur at values of x where f(x) = O, or where the polynomial intersects the x- axis. Recall that odd-degree polynomial functions have at least one real zero and even- degree polynomial functions have any number of real zeros. So, the minimum number of times that an odd - degree polynomial intersects the x- axis is 1, and the minimum number of times that an even- degree polynomial intersects the x- axis is O