introduction algebraic expressions are mathematical statements that include numbers, operations, and...

IntroductionAlgebraic 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.

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5.1.1: 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

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.

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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 x is 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.

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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.

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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.

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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.

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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.

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Common Errors/Misconceptions• not following the order of operations • incorrectly identifying like terms • inaccurately combining terms involving subtraction • incorrectly combining terms by changing exponents

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Guided Practice

Example 1Identify 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.

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Guided Practice: Example 1, continued

1. Simplify the expression. The expression can be simplified by following the order of operations and combining like terms.

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6(x – 1) – x(3 – 2x) + 12 Original expression

6x – 6 – x(3 – 2x) + 12 Distribute 6 over x – 1.

6x – 6 – 3x + 2x2 + 12 Distribute –x over 3 – 2x.

3x + 6 + 2x2 Combine like terms: 6x and –3x; –6 and 12.

2x2 + 3x + 6 Rearrange terms so the powers are in descending order.


2. Identify all terms. There are three terms in the expression: 2x2, 3x, and 6.

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3. 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.

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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.

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5. Classify the expression as a monomial, binomial, or trinomial. The polynomial is a trinomial because it has three terms.

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6. 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.

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Guided Practice

Example 3A 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?

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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 2x2 + 15x.18


P = 2x2 + (3x + 1) + (3x + 2) + 4x + (5x – 3)Set up the equation using the given expressions.

P = 2x2 + 3x + 3x + 4x + 5x + 1 + 2 – 3 Reorder like terms.

P = 2x2 + 15x Combine like terms.


2. Identify all terms. There are two terms in this expression: 2x2 and 15x.

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3. 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.

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4. Identify any constants. Every number in the expression is multiplied by a variable; therefore, there is no constant.

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5. 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.

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introduction algebraic expressions are mathematical statements that include numbers, operations, and...

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