كتاب الطالب Integrated III الرياضيات للصف 10 منهج انجليزي Reveal الفصل 3 2021 – 2022

كتاب الطالب Integrated III الرياضيات للصف 10 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