How to Help With Factoring Polynomials in Math
Mathematicians use math in many different applications and for many different purposes. Algebra is the foundation of all higher-level mathematics and factoring polynomials is one of the key concepts of algebra. Many first- and second-year algebra students have trouble with factoring polynomials, but a little help from an experienced tutor can go a long way. By learning the basics of polynomial factoring, you can help someone who finds the process difficult and make the road to more complicated algebraic ideas that much easier.
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Instructions
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1
Go over the rules of exponents and make sure the student understands how to use them. Some common exponent rules useful in factoring polynomials include:
p^Q *p^b = p^(Q+R)
p^Q / p^R = p^Q-R
(p^Q)^R = p^Q*R
p^0 =1
Use the "p's" as the unknown variable and "Q" and "R" as real numbers. For example:
p^5*p^6 = p^30
p^5 / p^3 = p^2
(p^7)^5 = p^35
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2
Use the exponent rules and work with the student on basic factoring. Look at the polynomial for any numbers or variables that appear in each term. For example, the polynomial
5p^ 4 + 10p^ 8+15 p^ 12
contains the common factor 5p^4 in each term. Pull out the common factor in front of the equation by using the division rule for exponents. For example,
5p^ 4 + 10p^ 8+15 p^ 12
= 5p^4(1+2p^ 4+3p^ 8).
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3
Work with the student on factoring the difference of two perfect squares. When you multiply a number by itself, it yields a perfect square. When a perfect square is subtracted from a squared variable (for example, p^2), a difference of squares is formed and can be factored in a special way. To factor a difference of squares, use the formula
p^2 -- Q^2 = (p- Q)*(p+Q)
where Q is any real number and "p" is the unknown variable. For example,
p^2 -25 = (p-5)*(p+5).
According to "Intermediate Algebra Tutorial 28: Factoring Trinomials," from West Texas A&M University, "the sum of two squares DOES NOT factor."(See Reference 1) For example, you cannot factor the equation
p^2 +25.
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4
Make sure the student can factor perfect cubes. Like perfect squares, a perfect cube results from multiplying a number by itself and multiplying the result by the original number again (for example 5*5*5 =125). Unlike perfect squares, you can factor the sum of a cubed variable, p^3, and a perfectly cubed real number, 125, as well as the difference of a cubed variable and a perfect cube. Each instance requires a special equation. Make sure the student memorizes both of the equations:
(p^3 -- Q^3) = (p-Q)*(p^2 +Qp + Q^2)
(p^3 + Q^3) = (p+Q)*(p^2 -- Qp + Q^2)
For example,
p^3 + 125 = (p+5)*(p^2 -- 5p + 25,)
and the polynomial
p^3 -- 125 = (p+5)*(p^2 + 5p + 25).
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5
Show the student how to factor by grouping. In grouping, the student splits the middle term into two terms that will factor with the first and last terms of the polynomial and produce a common factor. For example the following procedure uses group factoring:
p^2 +8p+15
= p^2 +5p+3p+15
= (p^2+5p)+(3p+15)
= p*(p+5)+3*(p+5)
= (p+3)*(p+5)
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6
Work with polynomials in which the first term is not equal to one. The simplest polynomials are of the form
1p^2 + Qp + R
where Q and R are real numbers ,and "p" is the unknown variable, but the student should learn to factor polynomials when the first term is not equal to one. For example, divide the polynomial
3p^2-19p+20
into two binomials
(3p- Q)*(p-R)
Use trial and error to find the unknowns, "Q" and "R," of the two binomials. In this example, the student would look for two numbers where
(-Q)*(-R) = 20
and
3p*(-R) +1p*(-Q) = -19p.
Solve the polynomial with trial and error. The trial-and-error process can seem tedious at first, but, eventually, it gets easier. In this case you can factor the polynomial
3p^2-19p+20
into the factors
(3p - 4)*(p - 5)
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7
Look for polynomials with no factors in common. If all attempts at factoring the polynomial fail, then you may not be able to factor it.
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Tips & Warnings
Many polynomials use "A," "B," and "C" in the place of "Q," "R," and "S" for real numbers.
Likewise, "x" or "y" is commonly used in the place of "p," but you can use any letter you wish.
References
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