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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