Factorising: (x+y)^2 - x^2, x^5 - 81x, a^2 +3ab +2b^2

[FlightVelocity]

Hi there!

I'm really stuck about how to fully factorise the following equations and would appreciate some guidance as to how it's done.

(x+y)^2 - x^2

x^5 - 81x

a^2 +3ab +2b^2

Additionally, I have no clue how to do the following question:

"A rectangular metal plate measuring 10 centimetres has two squares cut from it, one of side "x"cm and the other of side 4cm. Show that the remaining area of metal, A, can be expressed as A = (8-x)(8+x) square centimetres.

Once again thanks for your help.

 

[Harry_the_cat]

Hi there!

I'm really stuck about how to fully factorise the following equations and would appreciate some guidance as to how it's done.

(x+y)^2 - x^2

x^5 - 81x

a^2 +3ab +2b^2

Additionally, I have no clue how to do the following question:

"A rectangular metal plate measuring 10 centimetres has two squares cut from it, one of side "x"cm and the other of side 4cm. Show that the remaining area of metal, A, can be expressed as A = (8-x)(8+x) square centimetres.

Once again thanks for your help.

Q1. You need to use the "difference of two squares" rule: \(\displaystyle a^2 - b^2 = (a+b)(a-b)\).
In this case let x+y=a and x=b. Apply the rule and see what you get.

Q2. Always the first thing to do when factorizing is to take out any common factors. Can you see that the common factor here is x? Factorise it out and what have you got left? Hint: then use the diff of two squares rule to factorise further - you'll need to use the rule twice.

Q3. Split the middle term: \(\displaystyle a^2 + 3ab + 2b^2 =a^2 +ab + 2ab + 2b^2\). Now how do you normally factorise an expression with 4 terms?

Q4. Is there a misprint in your question? Is the other of side 6cm not 4cm? Otherwise the question can't be done.

 

[mmm4444bot]

A rectangular metal plate measuring 8 by 10 centimetres has two squares cut from it, one of side "x"cm and the other of side 4cm. Show that the remaining area of metal, A, can be expressed as A = (8-x)(8+x) square centimetres.

I've added a missing dimension, to the exercise statement. :cool:

Calculate the area of the rectangular plate.

Calculate the area of the squares.

Subtract the area of the squares from the area of the rectangular plate.

Factor the result.