Polynomials, Perfect Square, Differences in Squares and Cubes

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Introduction

There are 3 formula to remember for this page

Difference in squares

A² - B² =  (A+B)(A-B)

Product of cubes

A³ + B³ = (A + B)(A² - AB + B²) 

Difference in cubes

A³ - B³ = (A-B)(A² + AB +B²)

The Perfect Square

This was the bane of my life today and hopeful will be clear as day from now on. The perfect square trinomial is of the form

A² + 2AB + B² = (A+B)²

And this is the bit I needed. If you take square root term 1, and term 3 and multiply them by 2 and it equals term 2 you are golden.

So for example

x² + 8x + 16

Using the approach from above rather than perfect square, by grouping we can see factors of 16 which = 8 are 4,4 and so

(x+4)(x+4) = (x+4)²

Using the perfect square approach we do

Square root of x² = x
Square root of 16 = 4
Middle term is 2(4x)

Not a Perfect Square

So here is one which is not. This was the tutors step by step

9x² + 6x + 4

Square root of 9x² = 3
Square root of 4   = 2
3x2 = 6 but we have to double it because it is 2(AB) = 12 so not a perfect square

Is a Perfect Square

And a good one

4x² + 12x + 9

Square root of 4x² = 2
Square root of 9   = 3
3x2 = 6 but we have to double it because it is 2(AB) = 12 so is a perfect square
Looking what makes 4x² is 2x and what makes 9 is 3 so answer is
(2x+3)²

Difference in Squares

So if we want to do difference in square A²-B² = (A+B)(A-B). Needed to understand the (A+B)(A-B) so

 = (A+B)(A-B)
 = A² - AB + BA + B² 

Now

 -AB + AB = 0

So

 = A² - B² 

So looking at some examples

x² - 25 = (x+5)(x-5)
x² - 36 = (x+6)(x-6)
4x² - 9 = (2x+3)(2x-3)

And more complex 3x² -48 = 3(x²-16) = 3(x+4)(x-4) So they went to have this example 16x⁴ - 81y⁸ So this look straight forward and is but (4x² + 9y⁴)(4x² - 9y⁴) But the important thing to note is only difference is squares and use this approach so only right hand side (4x² + 9y⁴)(2x - 3y²)(2x + 3y²)

A really big one to enforce by knowledge

  4x² + 20x + 25 - (9y² - 24y - 16)
= 4x² + 20x + 25 = (2x+5)(2x-5)
= 9y² - 24y - 16 = (3y +4)(3y -4)

So using substitution we have

= [(2x+5)(2x-5)] - [(3y +4)(3y -4)]
= (2x+5)² - (3y +4)² // Difference of two squares
  A = 2x+5, B = 3y +4
= [(2x+5) + (3y +4)][(2x+5) - (3y +4)]
= [2x+3y+9][2x-3y+1]

Sum of Cubes

So off we go again A³ + B³ = (A + B)(A² - AB + B²) So given

  x³ + 8
= x³ + 2³  =(x+2)(x² - 2x + 4)

And another

  216x¹² + 343y¹⁵
   6³x¹² + 7³y¹⁵  A = 6x⁴, B = 7y⁵
  (6x⁴ + 7y⁵)(36x⁸ - (6x⁴)(7y⁵) + 49y¹⁰)
  (6x⁴ + 7y⁵)(36x⁸ - 42x⁴y⁵ + 49y¹⁰)

Difference of Cubes

So off we go again again A³ - B³ = (A - B)(A² + AB + B²) So given

  8x³ - 27  
= 8x³ - 3³  so A = 2x, B = 3
= (2x - 3)(4x² + 6x + 9)

And next

  x³ - 1/8
= x³ - 1/2³ so A = x, B = 1/2
= (x - 1/2)(x² + 1/2x +  1/4)

And a big one (This took a while to get the powers right - Divide exponents to reduce)

  125x⁶ - 64y⁹
= 5³x⁶ - 4³y⁶ A = 5x², B = 4y³
= (5x² - 4y²)(25x⁴ + (5x²)(4y²) + 16y⁴)
= (5x² - 4y³)(25x⁴ + 20x²y³ + 16y⁶)