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 : Identifying Terms, Factors, and Coefficients

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 ax 2 + 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 : Identifying Terms, Factors, and Coefficients

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

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: 4x 2, 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 4x 2 are 4 and x 2 and the factors of 6x are 6 and x : Identifying Terms, Factors, and Coefficients

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 4x 2 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 –x 2 means –1x 2. 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 : Identifying Terms, Factors, and Coefficients

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. 4x 2 + 6x – 2 has no like terms, so let’s use another expression as an example: 9x 2 – 8x 2 + 2x. In the expression 9x 2 – 8x 2 + 2x, 9x 2 and –8x 2 are like terms. After simplifying the expression by combining like terms (9x 2 and –8x 2 ), the result is x 2 + 2x : Identifying Terms, Factors, and Coefficients

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. 5x 2 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. 4x 2 + 6x – 2 is an example of a trinomial : Identifying Terms, Factors, and Coefficients

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

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

Guided Practice: Example 1, continued 1.Simplify the expression. The expression can be simplified by following the order of operations and combining like terms : Identifying Terms, Factors, and Coefficients 6(x – 1) – x(3 – 2x) + 12Original expression 6x – 6 – x(3 – 2x) + 12Distribute 6 over x – 1. 6x – 6 – 3x + 2x Distribute –x over 3 – 2x. 3x x 2 Combine like terms: 6x and –3x; –6 and 12. 2x 2 + 3x + 6 Rearrange terms so the powers are in descending order.

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

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

Guided Practice: Example 1, continued 4.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 : Identifying Terms, Factors, and Coefficients

Guided Practice: Example 1, continued 5.Classify the expression as a monomial, binomial, or trinomial. The polynomial is a trinomial because it has three terms : Identifying Terms, Factors, and Coefficients

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

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

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 2x 2, 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? : Identifying Terms, Factors, and Coefficients

Guided Practice: Example 3, continued 1.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 2x x : Identifying Terms, Factors, and Coefficients P = 2x 2 + (3x + 1) + (3x + 2) + 4x + (5x – 3) Set up the equation using the given expressions. P = 2x 2 + 3x + 3x + 4x + 5x – 3Reorder like terms. P = 2x xCombine like terms.

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

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

Guided Practice: Example 3, continued 4.Identify any constants. Every number in the expression is multiplied by a variable; therefore, there is no constant : Identifying Terms, Factors, and Coefficients

Guided Practice: Example 3, continued 5.Determine whether the expression is a quadratic expression. It is a quadratic expression because it can be written in the form ax 2 + bx + c, where a = 2, b = 15, and c = : Identifying Terms, Factors, and Coefficients ✔

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