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Introduction

Introduction

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Introduction

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  1. Introduction Algebraic expressions are mathematical statements that include numbers, operations, and variables to represent a number or quantity. We know that a variable is a letter used to represent a value or unknown quantity that can change or vary. We have seen several linear expressions such as 2x + 1. In this example, the highest power of the variable x is the first power. In this lesson, we will look at expressions where the highest power of the variable is 2. 5.1.1: Identifying Terms, Factors, and Coefficients

  2. Key Concepts A quadratic expression is an expression where the highest power of the variable is the second power. A quadratic expression can be written in the form ax2+ bx + c, where x is the variable, and a, b, and c are constants. Both b and c can be any number, but a cannot be equal to 0 because quadratic expressions must contain a squared term. 5.1.1: Identifying Terms, Factors, and Coefficients

  3. Key Concepts, continued An example of a quadratic expression is 4x2 + 6x – 2. When a quadratic expression is set equal to 0, as in 4x2+ 6x – 2 = 0, the resulting equation is called a quadratic equation. A quadratic equation is an equation that can be written in the form ax2+ bx + c = 0, where xis the variable, a, b, and c are constants, and a ≠ 0. The quadratic expression 4x2 + 6x – 2 is made up of many component parts: terms, factors, coefficients, and constants. 5.1.1: Identifying Terms, Factors, and Coefficients

  4. Key Concepts, continued A term is a number, a variable, or the product of a number and variable(s). There are 3 terms in the given expression: 4x2, 6x, and –2. A factor is one of two or more numbers or expressions that when multiplied produce a given product. In the given expression, the factors of 4x2 are 4 and x2 and the factors of 6x are 6 and x. 5.1.1: Identifying Terms, Factors, and Coefficients

  5. Key Concepts, continued The number multiplied by a variable in an algebraic expression is called a coefficient. In the given expression, the coefficient of the term 4x2 is 4 and the coefficient of the term 6x is 6. When there is no number before a variable, the coefficient is either 1 or –1 because x means 1x and –x2 means –1x2. A term that does not contain a variable is called a constant term because the value of the term does not change. In the given expression, –2 is a constant. 5.1.1: Identifying Terms, Factors, and Coefficients

  6. Key Concepts, continued Two or more terms that contain the same variables raised to the same power are called like terms. Like terms can be combined by adding. Be sure to follow the order of operations when combining like terms. 4x2+ 6x – 2 has no like terms, so let’s use another expression as an example: 9x2 – 8x2 + 2x. In the expression 9x2 – 8x2 + 2x, 9x2 and –8x2 are like terms. After simplifying the expression by combining like terms (9x2 and –8x2), the result is x2 + 2x. 5.1.1: Identifying Terms, Factors, and Coefficients

  7. Key Concepts, continued A monomial is a number, a variable, or the product of a number and variable(s). We can also think of a monomial as an expression containing only one term. 5x2 is an example of a monomial. A polynomial is a monomial or the sum of monomials. A polynomial can have any number of terms. A binomial is a polynomial with two terms. 6x + 9 is an example of a binomial. A trinomial is a polynomial with three terms. 4x2+ 6x – 2 is an example of a trinomial. 5.1.1: Identifying Terms, Factors, and Coefficients

  8. Common Errors/Misconceptions not following the order of operations incorrectly identifying like terms inaccurately combining terms involving subtraction incorrectly combining terms by changing exponents 5.1.1: Identifying Terms, Factors, and Coefficients

  9. Guided Practice Example 1 Identify each term, coefficient, and constant of 6(x – 1) – x(3 – 2x) + 12. Classify the expression as a monomial, binomial, or trinomial. Determine whether it is a quadratic expression. 5.1.1: Identifying Terms, Factors, and Coefficients

  10. Guided Practice: Example 1, continued Simplify the expression. The expression can be simplified by following the order of operations and combining like terms. 5.1.1: Identifying Terms, Factors, and Coefficients

  11. Guided Practice: Example 1, continued Identify all terms. There are three terms in the expression: 2x2, 3x, and 6. 5.1.1: Identifying Terms, Factors, and Coefficients

  12. Guided Practice: Example 1, continued Identify all coefficients. The number multiplied by a variable in the term 2x2 is 2; the number multiplied by a variable in the term 3x is 3; therefore, the coefficients are 2 and 3. 5.1.1: Identifying Terms, Factors, and Coefficients

  13. Guided Practice: Example 1, continued Identify any constants. The quantity that does not change (is not multiplied by a variable) in the expression is 6; therefore, 6 is a constant. 5.1.1: Identifying Terms, Factors, and Coefficients

  14. Guided Practice: Example 1, continued Classify the expression as a monomial, binomial, or trinomial. The polynomial is a trinomial because it has three terms. 5.1.1: Identifying Terms, Factors, and Coefficients

  15. Guided Practice: Example 1, continued Determine whether the expression is a quadratic expression. It is a quadratic expression because it can be written in the form ax2 + bx + c, where a = 2, b = 3, and c = 6. ✔ 5.1.1: Identifying Terms, Factors, and Coefficients

  16. Guided Practice: Example 1, continued 5.1.1: Identifying Terms, Factors, and Coefficients

  17. Guided Practice Example 3 A fence surrounds a park in the shape of a pentagon. The side lengths of the park in feet are given by the expressions 2x2, 3x + 1, 3x + 2, 4x, and 5x – 3. Find an expression for the perimeter of the park. Identify the terms, coefficients, and constant in your expression. Is the expression quadratic? 5.1.1: Identifying Terms, Factors, and Coefficients

  18. Guided Practice: Example 3, continued Find an expression for the perimeter of the park. Add like terms to find the perimeter, P. The expression for the park’s perimeter is 2x2 + 15x. 5.1.1: Identifying Terms, Factors, and Coefficients

  19. Guided Practice: Example 3, continued Identify all terms. There are two terms in this expression: 2x2 and 15x. 5.1.1: Identifying Terms, Factors, and Coefficients

  20. Guided Practice: Example 3, continued Identify all coefficients. The number multiplied by a variable in the term 2x2 is 2; the number multiplied by a variable in the term 15x is 15; therefore, 2 and 15 are coefficients. 5.1.1: Identifying Terms, Factors, and Coefficients

  21. Guided Practice: Example 3, continued Identify any constants. Every number in the expression is multiplied by a variable; therefore, there is no constant. 5.1.1: Identifying Terms, Factors, and Coefficients

  22. Guided Practice: Example 3, continued Determine whether the expression is a quadratic expression. It is a quadratic expression because it can be written in the form ax2 + bx + c, where a = 2, b = 15, and c= 0. ✔ 5.1.1: Identifying Terms, Factors, and Coefficients

  23. Guided Practice: Example 3, continued 5.1.1: Identifying Terms, Factors, and Coefficients

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