Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

 Topic Overview: Part 1: Factoring Quadratic Expressions Part 2: Understanding Basic Recursion

Factoring is the process of rewriting a number or expression as a product of two or more numbers or expressions. It can be used to break a polynomial into smaller parts. By writing the factors of a polynomial, it is often easier to solve equations. The distributive property plays a big role when multiplying factors to get a product.

Explanation

A factor (of a polynomial) is a polynomial that, when multiplied by another polynomial, yields the original polynomial. For instance, x and (x - 3) are factors of x2 - 3x because x(x - 3) = x2 - 3x. Similarly, (x - 5) and (x + 2) are factors of x2 - 3x - 10, because (x - 5)(x + 2) = x2 - 3x - 10.

When used as a verb, factor means to divide a number or expression into a product of other numbers or expressions. As an example, the number 30 can be factored as 1 x 30, 2 x 15, 3 x 10, 5 x 6, or 2 x 3 x 5. The last set of factors is called the prime factorization of 30 because the factors are prime numbers. On the other hand, there is typically only one way to factor a polynomial, especially when the leading coefficient of x2 is 1. For instance, the expression x2 + 2x - 24 can only be factored as (x + 6)(x - 4).

Many polynomials, such as x2 + 2x - 7, cannot be factored at all, and are therefore known as prime polynomials. Some polynomials, however, contain coefficients with common factors, and they may be factored in more than one way. For instance, 2x2 + 4x - 48 can be factored in several different ways: 2(x + 6)(x - 4) or (2x + 12)(x - 4) or (x + 6)(2x - 8).

Factoring is one method of finding the solutions of a polynomial equation. Using factoring, the quadratic equation x2 + 2x - 15 = 0 can be rewritten as (x + 5)(x - 3) = 0. This shows that either x + 5 = 0 or x - 3 = 0, because one or both of the factors must equal zero if their product is to equal zero. Therefore, either x = -5 or x = 3.

Other examples:
• The height (h) of a ball thrown into the air with an initial vertical velocity of 24 feet per second from a height of 6 feet above the ground is represented by the equation h = 16t2 + 24t + 6 where t is the time, in seconds, that the ball has been in the air. After how many seconds is the ball at a height of 14 feet?

• The parabola y = x2 + 18x - 40 crosses the x-axis at (2, 0) and (-20, 0); the function can be factored as y = (x + 20)(x - 2).

• A garden that is 20 feet by 30 feet is bordered by a cement walkway. If the width of the walkway is w feet, the area of the garden and walkway combined is (20 + 2w)(30+2w) = 600 + 100w+ 4w2 or (2w+20)(2w+30) = 4w2 + 100w + 600

• A rectangular box has a volume of 280 in3. Its dimensions are 4 in * (x + 2) in * (x + 5) in. The formula V = lwh can be used to find the value of x, as follows:
4(x + 2)(x + 5) = 280
4x2 + 28x + 40 = 280
4x2 + 28x - 240 = 0
x2 + 7x - 60 = 0
(x + 12)(x - 5) = 0
x = -12 or x = 5
Because x = -12 does not make sense in the context of the problem, it cannot be an answer. Consequently, x = 5, and the dimensions are
4 in, 7 in, and 10 in.

• The equation -0.00239d2 + 1.199d = 0 can be used to model a home run that Mickey Mantle hit on May 22, 1963, where d represents the distance of the home run in feet. By factoring the expression as 0.00239d (d - 501.6736) = 0, students can see that Mantle's hit measured about 502 feet.
Mathematical Definition

A factor (of a number) is an integer that, when multiplied by another integer, results in the original number. For instance, 3 is a factor of 18, because 3 x 6 = 18, and a and b are factors of ab.

A factor (of a polynomial) is a polynomial that, when multiplied by another polynomial, results in the original polynomial. For instance, (x + 2) and (x + 4) are factors of x2 + 6x + 8, because (x + 2)(x + 4) = x2 + 6x + 8. Similarly, 4 and (x + 5) are factors of 4x + 20 because 4(x + 5) = 4x + 20.

To factor a number (or a polynomial) is to name the number by its factors; that is, to write the number (or polynomial) as a product of two or more integers (or polynomials). When used as a verb, factoring refers to the process opposite of multiplying. In a sense, it is similar to dividing - to factor a number or polynomial is to divide it into other numbers or polynomials.

Following is a definition of factor from a mathematics dictionary:

Factor: As a verb, to resolve into factors. One factors 6 when he writes it in the form 2 x 3. [As a noun,] a factor of an object (perhaps of some specified type) divides the given object.
factor of an integer: An integer whose product with some integer is the given integer. For example, 3 and 4 are factors of 12, since 3 x 4 = 12; the positive factors of 12 are 1, 2, 3, 4, 6, 12, and the negative factors are -1, -2, -3, -4, -6, -12.

factor of a polynomial: One of two or more polynomials whose product is the given polynomial. Sometimes one of the polynomials is not allowed to be the constant 1, but usually in elementary algebra a polynomial with rational coefficients is considered factorable if and only if it has two or more nonconstant polynomial factors whose coefficients are rational (sometimes it is required that the coefficients be integers). For example, (x2 - y2) has the factors (x - y) and (x + y) in the ordinary (elementary) sense; (x2 - 2y2) has the factors and in the field of real numbers; (x2 + y2) has the factors (x - iy) and (x + iy) in the complex field.
(Source: James, Robert C. and Glenn James. Mathematics Dictionary (5th edition). New York: Chapman & Hall, 1992)

Role in the Curriculum

Exposure to factoring is important for Algebra 1 students because it illuminates many aspects of the nature of mathematics and serves as a bridge to the study of advanced mathematical topics. The key word here is exposure. In the past, students would be entrenched in the study of factoring for an entire unit and spend a month learning myriad methods; today, the topic receives less emphasis. Students should understand the various representations for polynomials, one of which is its factored form, rather than mastering every factoring technique.

Read what teacher educator Diane Briars has to say about the role of factoring in the curriculum:

 Read transcript from teacher educator Diane Briars In terms of solving quadratics, we need to recognize that factoring is limited as a general method... Read More

The NCTM Principles and Standards for School Mathematics (PSSM) does not explicitly mention factoring polynomials. However, the Algebra Standard lists the following expectation: "Students should understand relations and functions and select, convert flexibly among, and use various representations for them; ... [and] understand the meaning of equivalent forms of expressions, equations, inequalities, and relations." (PSSM 2000, p. 395)

Recognizing equivalent forms of expressions and being able to convert flexibly among them means that a student should be able to write a polynomial in factored form. That is, a student should understand that x2 + 7x + 10 = (x + 2)(x + 5). Further, students should recognize that both expressions represent a quadratic function that crosses the x-intercept at
(2, 0) and (-5, 0).

According to Workshop 5 video teacher Tom Reardon, factoring trinomials gives his algebra students their first exposure to quadratic functions and lays the foundation for in-depth study of quadratics later in the year. Students should be able to factor expressions by graphing the equivalent function and understanding the relationship between the x-intercepts and the factors. The strength of the graphical technique is that it is generalized to all polynomials and not just quadratic expressions. Computer Algebra Systems (CAS) can also assist in factoring complicated expressions. (CAS manipulate a formula symbolically using the computer. Factoring, finding roots, and simplifying polynomials are some of the typical uses of CAS.) Students should be able to factor some expressions mentally, such as x2 + 2x + 1, while more complex expressions, such as 3.2x2 + 7x - ½, should be solved graphically or with CAS. They should recognize when one technique is more efficient or beneficial than the others.

Read what teacher Tom Reardon has to say about the role of factoring in the algebra curriculum:

 Read transcript from teacher Tom Reardon It's a technique that's going to be used to solve quadratic equations later, or maybe help work with rational expressions... Read More