Objective: To recall how polynomials factor.
Recall 1: We recall that
x^2 + 5*x = x*(x + 5),
x^3 + 5x^2
+ 4x = x(x^2 + 5x + 4),
3x^3 + 6x^2 + 9x = 3x(x^2 + 2x + 3).
Recall 2: We recall that
(x + 5) (x - 5) = x^2 - 25,
and that, more generally,
(x + b)(x - b) = x^2 - b^2.
Recall 3: A slightly different form of this last recall is
(3x + 5)(3x - 5) = 9x^2 - 25,
or, more generally,
(ax + by) (ax - by) = (ax)^2 - (by)^2.
Notice that if a = 1, and y = 1, this reduces to the form in Recall 2.
Very often, a difference of two
squares is not recognized,
simply because it is not written in a
For example, 6x^3 - 150x^2 does not look like the difference of two squares. Once you factor out a copy of 6x from both terms, you have the expression 6x(x^2 - 25). The term in parentheses is the difference of two squares.
Recall 4: Look for all the products that repeatedly appear in each term of a polynomial expression. For example, the expression
5x^2*y^2 + 3xy^3 +xy^2
is the same as
5x(xy^2) + 3(xy^2)y + (xy^2).
Since an (xy^2) appears in each term, we may factor it out and rewrite it as
(xy^2)(5x + 3y +1).
Recall 5: Another product to recall is
(x - 1) (x^2 + x + 1) = x^3 - 1,
and that, more generally,
(x - b) (x^2 + bx + b^2) = x^3 - b^3.
The expression on the right is occasionally referred to as the difference between two cubes.
Very often, a difference of two cubes is not recognized, simply because it is not written in a recognizable form.
For example, 4x^4 - 108x does not look like the difference of two squares. Once you factor out a copy of 4x from both terms, you have the expression 4x(x^3 - 27). The term in parentheses is the difference of two cubes.
Example 1. Factor 55x^22 + 11x^11.
Solution: Notice that there is a copy of 11x^11 in 55x^22, since
55x^22 = (11x^11)(5x^11).
So the factored form of the given polynomial is 11x^11(5x^11 + 1).
Always look for the lowest common power of the variable that appears in all the terms of the polynomial.
Example 2. Factor 8x^3 - 200x.
Solution: There is a copy of 8x in each term. Notice that
8x^3 = (8x)x^2, and that
-200x = (8x)(-25).
Therefore, we may pull out a copy of 8x from each term to rewrite the original expression as
8x(x^2 - 25).
Now, notice that x^2 - 25 is the
two squares and may be factored
(x + 5)(x - 5);
so the original expression may be completely factored as
8x(x + 5)(x - 5).
Example 3. Factor x^2 + 24x + 143.
Solution: Look for the factors of 143.
143 = 1 *143,
= (-1) * (-143),
= 11 * 13,
= (-11) * (-13).
The only factors whose sum is +24 are 11 and 13; and so we have:
x^2 + 24x + 143 = (x + 11)(x + 24).
In general, to factor a polynomial such as x^2 + bx + c, look for products of the form
(x ± some number)*(x ± some number).
Sometines you may not be able to factor the polynomial in this way. We shall consider this situation in Workout 13.
Example 4 Factor 55x^100*y^50*z^25 + 110x^25*y^50*z^100.
Solution: Write 110 as 55 * 2, so the expression becomes
55(x^100*y^50*z^25 + 2*x^25*y^50*z^100).
Next, notice that the lowest common power of x in both terms is 25, the lowest common power of y in both terms is 50 and that the lowest common power of z in both terms is 25. So, we may factor out the product of all these lowest common powers, namely x^25*y^50*z^25. We may, therefore, rewrite the expression as
55x^25*y^50*z^25.(x^75 + 2z^75).
Example 5. Factor x^3 - 1331.
Solution: This is the difference of two cubes, since 11^3 = 1331. We may use the formula of the difference between two cubes:
x^3 - a^3 = (x - a)(x^2 + ax + a^2)
In the case of this example, a^3 = 1331, so a = 11. Therefore,
x^3 - 1331= (x - 11)(x^2 + 11x + 11^2)
= (x - 11)(x^2 + 11x + 121).
If the example above asked you to
factor x^3 + 1331, then
you could write it as x^3 - (-1331), which is the
same as x^3 - (-11)^3.
You would then have a factoring
x^3 + 1331= (x - (-11))(x^2 + (-11)x + (-11)^2)
= (x + 11)(x^2 - 11x + 121).
Example 6 Show that 6x^4 - 1296x = 6x(x - 6)(x^2 + 6x + 36).
Solution: Occasionally, differences of two cubes disguise themselves. In this case, we may remove the disguise by factoring out a copy of 6x from both terms to get
6x(x^3 - 216).
The remaining part of the demonstration follows from an argument similar to the previous example; i.e. that
(x^3 - 216) = (x - 6)(x^2 + 6x + 36), since 216 = 6^3.
Factor the following polynomials:
1. x^2 +
2. x^3 + x^2 +
3. 4x^3 - 3x^2 +
4. x^2 -
5. 4x^2 -
6. 3x^3 -
7. x^2 + 6x +
8. x^2 - 8x +
9. x^2 + 3x -
10. x^3 + 7x^2 +
11. x^3*y -
12. 5x^2y^2 + 3xy^3
13. sqrt(5)*x^3y +
14. 36 + 12*x +
15. x^3 -
16. x^4 - 27*x
x*(x + 1)
2. x*(x^2 + x +
3 x*(4*x^2 - 3*x +
4 (x - 6)*(x +
5 4*(x - 5)*(x +
6 3*x*(x - 5)*(x +
7. (x + 3)*(x + 3)
8 (x - 4)*(x - 4)
9 (x + 5)*(x - 2)
10. x*(x + 3)*(x + 4)
11 x*y*(x - y)*(x + y)
12 x*y^2*(5*x + 3*y + 1)
13 x^2*y*(*x + )
14 (x + 6)*(x + 6)
15 (x - 3)*(x^2 + 3*x + 9)
16 x*(x - 3)*(x^2 + 3*x + 9)