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Mathematics

Mathematics. Instructor : Brwa A. Saeed – MSc . Structural Engineering brwa.ahmed@ishik.edu.iq Faculty of Engineering Ishik University. Lecture – 3 / Introduction to Polynomials Solve System of Linear Equation with One Variable Fall Semester 2017. Ishik University.

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Mathematics

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  1. Mathematics Instructor: Brwa A. Saeed – MSc . Structural Engineering brwa.ahmed@ishik.edu.iq Faculty of Engineering Ishik University Lecture – 3 / Introduction to Polynomials Solve System of Linear Equation with One Variable Fall Semester 2017

  2. Ishik University Introduction to Polynomials • Monomial:A number, a variable or the product of a number and one or more variables. (axn with n is a non-negative integers, a is a real number) • Examples: 3x, -3, or 4xy2z • Polynomial:A monomial or a sum of monomials. • Binomial:A polynomial with exactly two terms. • Examples: 3x – 5 , or4xy2z + 3ab • Trinomial:A polynomial with exactly three terms. • Examples:4x2 + 2x - 3 Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  3. Ishik University Introduction to Polynomials Identify the parts of monomial Exponent Exponent Coefficient Coefficient Variable Variable Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  4. Ishik University Introduction to Polynomials • Coefficient: A numerical factor in a term of an algebraic expression, such as number 4 in term 4x. • Degree of a monomial:The sum of the exponents of all of the variables in the monomial. • Degree of a polynomial:is equal the highest exponent (if the term has more than 1 variable, then add all exponents of that term) • Standard form: When the terms of a polynomial are arranged from the largest exponent to the smallest exponent in decreasing order. Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  5. Ishik University Degree of Monomials • What is the degree of the monomial? • The degree of a monomial is the sum of the exponents of the variables in the monomial. • The exponents of each variable are 3 and 2. • The degree of the monomial is 5. • The monomial can be referred to as a fifth degree monomial. Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  6. Ishik University Degree of Monomials • A polynomial is a monomial or the sum of monomials • Examples:,, • Each monomial in a polynomial is a term of the polynomial. • The number factor of a term is called the coefficient. • The coefficient of the first term in a polynomial is the lead coefficient. • A polynomial with two terms is called a binomial. • A polynomial with three terms is called a trinomial. Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  7. Ishik University Degree of Monomials • The degree of a polynomial in one variable is the largest exponent of that variable. • 4 is a constant has no variable. It’s called a 0 degree polynomial. • This is a 1st degree polynomial. 1st degree polynomials are linear. • This is a 2nd degree polynomial. 2nd degree polynomials are quadratic. • This is a 3rd degree polynomial. 3rd degree polynomials are cubic. Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  8. Ishik University Degree of Monomials • Are these polynomials or not polynomials? Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  9. Ishik University Degree of Monomials Classify the polynomials by degree and number of terms. Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  10. Ishik University Standard Forms • To rewrite a polynomial in standard form, rearrange the terms of the polynomial starting with the largest degree term and ending with the lowest degree term. • The leading coefficient, the coefficient of the first term in a polynomial written in standard form, should be positive. Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  11. Ishik University Standard Forms Examples: Write the polynomials in standard forms Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  12. Ishik University Standard Forms Examples: Write the polynomials in standard forms Remember: The lead coefficient should be positivein standard form. To do this, multiply the polynomial by –1 using the distributive property. Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  13. Ishik University Standard Forms Examples: Write the polynomials in standard forms Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  14. Ishik University Solve Problems Examples: Write down one example of each of the following types of polynomial function: • Cubic • Linear • Quartic • Quadratic In Class Activity Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  15. Ishik University Solve Problems Write the polynomials in standard form and identify the polynomial by degree, number of terms and leading coefficient. Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  16. Ishik University Solve Problems • This is a 2nd degree polynomial (quadratic) with binomial terms • Leading Coefficient = 1 Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  17. Ishik University Solve Problems • This is a 5th degree polynomial with 5 terms • Leading Coefficient = 7 Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  18. Ishik University Solve Problems • This is a 9thdegree polynomial with 5 terms • Leading Coefficient = 6 Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  19. Ishik University Solve Problems • 3 • This is a 8thdegree polynomial with 4 terms • Leading Coefficient = 3 Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  20. Ishik University Linear Equations • Linear equations are first degree polynomial equations. • A linear equation with one variable can be written in the form of • Where, a, b and c are all real numbers and Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  21. Ishik University Left side Right side Equation (to solve) Linear Equations Distinguish between expressions and equations. • Equations and inequalities compare algebraic expressions. • An equation is a statement that two algebraic expressions are equal. • An equation always contains an equals symbol, while an expression does not. Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable 3x – 7 = 2 3x – 7 Expression (to simplify or evaluate)

  22. Ishik University Linear Equations with One Variable So, can you tell these equations are linear equation with one variable or not? NO, because there are 2 variables x and y NO, because is not a linear equation Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable NO, because there is variable at denominator

  23. Ishik University Linear Equations with One Variable Strategy for solving Linear Equations. Isolate variable terms on one side and constant terms on the other side. Problem-1. Solve the following Linear Equations. • 8 Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  24. Ishik University Solve Linear Equations with One Variable Example – 1: Solve the following Linear Equations. Check for x Check for x Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  25. Ishik University Solve Linear Equations with One Variable Example – 1: Solve the following Linear Equations. Check for x Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  26. Ishik University Solve Linear Equations with One Variable To generate equivalents equations you can use the following operations. • Simplify an expression by removing grouping symbols and combining like terms. • Add/subtract the same real number/expression on both sides of the equation. • Multiply/divide on both sides of the equation by the same nonzero quantity • Interchange two sides of the equation. Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  27. Ishik University Solve Linear Equations with One Variable Example – 2: Solve the following linear equation. Multiply number with inside parenthesis Put all variables at the left side and numbers at the right side. Add or subtract variables with same variables and number with number if there is any. Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable Multiply both side by -1 Divide both side by 2 to get the value of x

  28. Ishik University Solve Linear Equations with One Variable Example – 2: Solve the following linear equation. Check for x Substitute the value of x = 5.5 in the equation, if left side equals right side then value of x is correct Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  29. Ishik University Solve Linear Equations with One Variable Example – 3: Solve the following linear equation. Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  30. Ishik University Solve Linear Equations with One Variable Example – 4: Solve the following linear equation. Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  31. Identification of Linear Equation How to Determine the Solution Sets of Linear Equations

  32. Ishik University Identification of Linear Equation Identify conditional equations, contradictions, and identities. Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  33. Ishik University Identification of Linear Equation Example – 5: Solve each equation, Decide whether it is conditional, identity, or contradiction. So, this equation is Conditional And solution set = {-3} Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  34. Ishik University Identification of Linear Equation Example – 5: Solve each equation, Decide whether it is conditional, identity, or contradiction. So, this equation is Identity And solution set = {real numbers} Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  35. Ishik University Identification of Linear Equation Example – 5: Solve each equation, Decide whether it is conditional, identity, or contradiction. So, this equation is Contradiction And solution set = {} or no solution Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  36. Ishik University Identification of Linear Equation Example – 6: Solve each equation, Decide whether it is conditional, identity, or contradiction. Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable So, this equation is Conditional And solution set = {24}

  37. Ishik University Home Work Example – 7: Solve each equation, Decide whether it is conditional, identity, or contradiction. • 3 Faculty of Engineering – Math– Lecture 3 – Introduction to Polynomials – Solve System of Linear Equ. with one Variable

  38. THANKS I HOPE YOU HAVE ENJOYED MY CLASS

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