Quandaries and Queries
 

 

Who is asking: Student
Level: Secondary

Question:
Factor completely:
3x3 - 24y3
54x6 + 16y3
16xy - 4x - 4y - 1
0.09x2 - 0.16y2

 

 

Hi,

The essential of factoring is recognizing a pattern. Three of the main patterns you need to recognize are

a difference of squares
a2 - b2 = (a - b)(a + b)

a difference of cubes
a3 - b3 = (a - b)(a2 + ab + b2) , and

a sum of cubes
a3 + b3 = (a + b)(a2 - ab + b2)

To be able to apply these facts you need to be able to look at an algebraic expression and see

something squared minus something squared or

something cubed minus something cubed or

something cubed plus something cubed

For example in your first problem

3x3 - 24y3

there is a common factor of 3 so it can be written

3x3 - 24y3 = 3(x3 - 8y3).

Now I see a difference of cubes since 8 is 23 and hence

x3 - 23y3 = x3 - (2y)3

Now factor as a difference of cubes.

Problem 2 is similar. Take out a common factor and you will have a sum of cubes since x6 = (x2)3

Problem 4 is similar.

In problem 3, 16xy - 4x = 4x(4y - 1) so

16xy - 4x - 4y - 1 = 4x(4y - 1) - 4y -1

In this expression 4y - 1 is almost a common factor. If the expression were

4x(4y - 1) + 4y -1

then 4y - 1 would be a common factor. Do you have a typo?

Penny

 
 

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