This is a free lesson from our course in Algebra I
In this lesson you’ll learn about Factoring 3rd Degree Polynomials using
different possible ways. Now you will proceed further taking help of earlier learning
and familiarity of related terms used for factoring. Generally you can use grouping,
finding common factors and then other common factors or synthetic division.
You can also use long division of polynomials, covered else where. This lesson
will also help you learn important rules and how to make use of them. The
presentation covering such content will be done by the instructor in own handwriting,
using video and with the help of several examples with solution to take a look at
methods to do so. (More text below video...)
(Continued from above) Factoring
a third degree polynomial also called cubic polynomial involves: you should look first
for the Greatest Common Factor (GCF), next if any of the identities can be used
and finally attempt factoring by grouping. If yet to work further, you may use the
rational roots test and the factor theorem.
A 3rd degree polynomial can be expressed either as a product of three linear polynomials
or product of one linear polynomial and one quadratic polynomial. E.g. a_{0}x^{3}
+ a_{1}x^{2} + a_{2}x + a_{3}
= k(x – a) (x – b) (x – c) is the product of three linear
polynomials and a_{0}x^{3} + a_{1}x^{2}
+ a_{2}x + a_{3} = k(x – a) (Ax^{2} +
B x + C) is the product of one linear and one quadratic polynomial respectively.
Thus the 3^{rd} degree polynomial can be factored in one of the above forms
and in this lesson you’ll also learn how to do it.
To proceed, first identify if the given polynomial follows any relevant product pattern.
Notice that the important identities that may guide for factoring third degree polynomials
are:
• m^{3} + n^{3} = (m + n)(m^{2}  m n + n^{2})
• m^{3}  n^{3} = (m  n)(m^{2} + m n + n^{2})
• (m + n) ^{3} = m^{3} + n^{3} + 3mn(m + n) = m^{3}
+ n^{3} + 3m^{2} n + 3mn^{2}
• (m – n) ^{3} = m^{3} – n^{3} – 3mn(m – n) = m^{3}
– 3m^{2} n + 3mn^{2} – n^{3}
For example  Factorize the given 3rd degree polynomial: 27x^{3} – 1
Note that you can use the identity m^{3} n^{3} = (m  n)(m^{2}
+ mn + n^{2}) to solve it.
Factoring by taking out the GCF:
In such cases you can identify the factors of each term of the polynomial
and take out common factors, as possible. For example: Given polynomial is x
^{3} + 13 x ^{2}  30 x, and you are required
to do factoring.
Here you can apply basic factoring techniques to secondand simple thirddegree
polynomials. This may include finding a common factor for all terms in a polynomial,
recognizing the difference of two squares, and also recognizing perfect squares
of binomials. Say, first factor out x, and get = x(x^{2}
+ 13x  30). Then factor: x^{2} + 13x  30.
It can be bone ay finding numbers a, and b such that: x^{2} + 13x – 30 = (x a) (x b), and a*b = 30 and
a – b = 13 (the difference
since the constant 30 is less than 0).
As the x coefficient is greater than 0, the sign before the larger number
15 is positive. Thus you’ll have: x^{2} + 13x – 30 = (x2) (x+15). Therefore, x^{3} +13x^{2}  30x = x (x2) (x+15).
Taking another example of a given polynomial 2x^{3} + 4x^{2}
+ 2x, you’ll note that the common factor i.e. GCF is 2x
Hence taking out 2x, you get 2x^{3} + 4x ^{2}
+ 2x = 2x(x^{2} + 2x + 1).
Now do further factoring using the techniques to factorize a quadratic. Thus,
2x^{3} + 4x^{2} + 2x = 2x(x + 1) (x + 1) = 2x(x + 1) ^{2}
2x(x
+ 1)^{2} , as the final answer.
Factorizing by grouping: You have to first group the terms of the 3^{rd} degree polynomial.
Then take out the common factor in each group and use the technique of factoring
quadratics. For example:
Factorize given polynomial: x^{3} – 3x^{2} – 36x
+ 108
Grouping: x^{3} – 3x^{2} – 36x+ 108 = (x^{3} – 3x^{2} )+ (– 36x + 108)
Common factor: (x^{3} – 3x^{2}) + (– 36x + 108)
= x^{2} (x – 3) – 36(x – 3)
Common factor: x^{2} (x – 3) – 36(x – 3) = (x – 3)
(x^{2} – 36)
Quadratic factor: (x – 3) (x^{2} – 36) = (x – 3) (x – 6) (x
+ 6)
Thus, x^{3} – 3x^{2} – 36x + 108 = (x – 3) (x – 6) (x + 6)
Note: the specific technique factor resulting in the solution is indicated in ‘italics’.
Remember: special factor patterns formulas that may be explored to match the polynomial
in question may be from
• difference of perfect cubes: m^{3}  n^{3} = (m  n)(m^{2} + mn + n^{2})
• sum of perfect cubes: m^{3} + b^{3} = (m + b)( m^{2}  mb + b^{2} )
• difference of perfect squares: x^{2}  m^{2}
= (x + m)(x  m)
Note the other important patterns:
• polynomials of degree 0: The nonzero constants P(x) ? a. Note: P(x)
? 0 (the zero polynomial) is a polynomial but no degree is assigned to it.
• polynomials of degree 1: Linear polynomials P(x) = mx+n.
The
graph of a linear polynomial is a straight line.
• polynomials of degree 2: Quadratic polynomials P(x) = mx^{2}
+ nx + c.
The graph of a quadratic polynomial is a parabola which opens up if a >
0, down if a < 0.
• polynomials of degree 3: Cubic polynomials P(x) = mx^{3}
+ nx^{2} + cx + d.
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