# Factoring Homework: Problems and Solutions

• Andy111
In summary, the conversation is about factoring problems, specifically (x-1)^3 - (x+2)^3, 64x^3 - 27y^3, and 3ab - 20cd -15ac + 4bd. The conversation includes tips on how to solve these problems, using substitutions and factoring by grouping. The conversation also discusses the formula for x^3 - y^3, which can be used to factor expressions with a minus sign instead of a plus sign.
Andy111

## Homework Statement

I had a topic somewhere about factoring, and now I have some more factoring problems I don't understand.

such as:

1)(x-1)$$^{3}$$ - (x+2)$$^{3}$$

2)64x$$^{3}$$ - 27y$$^{3}$$

3)3ab - 20cd -15ac + 4bd

## The Attempt at a Solution

1. If I use substitutions, it may make things a little easier.

$$a=(x-1)$$
$$b=(x+2)$$

$$a^3-b^3=(a-b)(a^2+ab+b^2)$$

$$(x-1)^3-(x+2)^3=[(x-1)-(x+2)][(x-1)^2+(x-1)(x+2)+(x+2)^2]$$
$$=(x-1-x-2)(x^2-2x+1+x^2+x-2+x^2+4x+4)$$

Does this confuse you more? Continue simplifying and collecting like terms and it's solved.

2. Can you re-write it?

What number must you raise to the power of 3, to attain 64 and 27? You want to choose a number so you can raise both your coefficient and variable to the same power.

3. Factor by grouping.

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No, that's not confusing, I understand what you're doing in 1.

I know 4 cubed and 3 cubed = 64 and 27, but I don't know how it looks in factored form.

Then you should understand that rocophyics just told you what it looks like in factored form: x3- y3= (x- y)(x2+ xy+ y2).

Last edited by a moderator:
Andy111 said:
I know 4 cubed and 3 cubed = 64 and 27, but I don't know how it looks in factored form.

Hi Andy!

64x$$^{3}$$ - 27y$$^{3}$$ = 4$$^{3}$$x$$^{3}$$ - 3$$^{3}$$y$$^{3}$$: does that help?

$$a^mb^m=(ab)^m$$

Oh, okay, so 2 is like 1, but with coefficients.

You've got 1) and 2) now.

Have you got 3)?

If not, there are various ways of doing it - one is to write it as a 2x2 matrix.

[size=-2](if you're ok now, don't forget to mark thread "solved"!)[/size]​

Yeah, I got 3,

I believe this is right (3a + 4d)(b - 5c).

I understand the formula for x^3 - y^3, but what the minus sign is instead a plus sign?

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$$(x+y)(x^2-xy+y^2)$$

Replace y, with (-y) and tell me what your new equation is.

factoring x^3 + y^3 ?

Andy111 said:
Yeah, I got 3,

I believe this is right (3a + 4d)(b - 5c).

Yay!

I understand the formula for x^3 - y^3, but what the minus sign is instead a plus sign?

Hint: divide by y^3, so you get (x/y)^3 + 1.

Put x/y = z, so you get z^3 + 1.

Can you see how to factor that?

(if not, come back for another hint)

Andy111 said:
Yeah, I got 3,

I believe this is right (3a + 4d)(b - 5c).

I understand the formula for x^3 - y^3, but what the minus sign is instead a plus sign?
$$x^n- y^n= (x- y)(x^{n-1}+ x^{n-2}y+ x^{n-3}y^2\cdot\cdot\cdot+ x^2y^{n-3}+ xy^{n-2}+ y^{n-1}$$
for n any positive integer.

$$x^n+ y^n= (x+ y)(x^{n-1}- x^{n-2}y+ x^{n-2}y^2\cdot\cdot\cdot- x^2y^{n-3}+ xy^{n-2}- y^{n-1}$$
for n any odd integer.

## What is factoring?

Factoring is the process of finding the factors of a given number or algebraic expression. Factors are numbers or expressions that when multiplied together, result in the original number or expression.

## Why is factoring important?

Factoring is important because it helps us simplify complex expressions, solve equations, and find the greatest common factor and least common multiple of numbers. It is also a crucial skill in algebra and other higher-level math courses.

## What are some common strategies for factoring?

Some common strategies for factoring include finding the greatest common factor (GCF), using the difference of squares, and using the grouping method. Other techniques include the quadratic formula and completing the square.

## What are the most common mistakes students make when factoring?

One common mistake students make when factoring is forgetting to check for a common factor before using other methods. Another mistake is not fully understanding the difference of squares or grouping method. It is also important to carefully check the signs and terms while factoring.

## How can I improve my factoring skills?

The best way to improve your factoring skills is through practice. Start with simpler problems and gradually increase the difficulty. It is also helpful to review the common factoring strategies and make sure you understand the concepts behind them. Additionally, seeking help from a teacher or tutor can also be beneficial.

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