This week one thing we learned was the different products that different equations can reach, one fully positive, one fully negative, and one both.

The distribution of the following polynomials

(a+b)^2 = (a+b)(a+b)

= a^2 + ab + ba + b^2

= a^2 +2ab + b^2

 

(a - b)^2 = (a – b)(a – b)

= a^2 – ab – ba + b^2

= a^2 – 2ab + b^2

 

(a – b)(a + b)

= a^2 + ab – ba – b^24

= a^2 – b^2

Knowing these patterns assist in longer polynomial expressions as we recognize the patterns allowing us to simplify quicker and easier, because we can go basically to the answer without needing to complete distributive property

For example

Expand and simplify (x + 4)(x – 4) – (x + 8)(x+8)

Because we know the terms are the same we do not need to go through distributive property and skip straight from (x + 4)(x – 4)  to  x^2 – 16

Continuing on, none of the product expression up top relate to (x + 2)(x+8) because they do not share all the same numbers so we need to answer them ourselves.

= x^2 – 16 – (x^2 + 8x + 2x + 16)

= x^2 – 16 – (x^2 + 10x + 16)

Bring in the negative.

= x^2 – 16 – x^2 – 10x – 16

The x^2 cancel out because there is a negative and a positive one of the same thing. Just finish by simplifying and adding like terms.

= -10x – 32