Question

100 Points!!!

Polynomial Identities

Part 1. Pick a two-digit number greater than 25. Rewrite your two-digit number as a difference of two numbers. Show how to use the identity (x − y)2 = x2 − 2xy + y2 to square your number without using a calculator.

Part 2. Choose two values, a and b, each between 8 and 15. Show how to use the identity a3 + b3 = (a + b)(a2 − ab + b2) to calculate the sum of the cubes of your numbers without using a calculator.

Answer 1

Let's see

#a

Take 28

• (28)²
• (30-2)²
• 30²-2(30)(2)+2²
• 900-120+4
• 780+4
• 784

#2

Take 9,10

• 9³+10³
• (9+10)(9²-9×10+10²)
• (19)(81-90+100)
• 19(181-90)
• 19(91)
• 1729

Answer 2

Two-digit number greater than 25: 32

Rewrite 32 as the difference of 2 numbers: 40 - 8

Therefore, x = 40 and y = 8

`\begin{aligned}\implies (40-8)^2 & =40^2-2(40)(8)+8^2\\ & = (4 \cdot 10)^2-(80)(8)+64\\ & = 4^2 \cdot 10^2-640+64\\ & = 16 \cdot 100-640+64\\ & = 1600-640+64\\ & = 960+64\\ & = 1024\end{aligned}`

Let a = 10

Let b = 11

`\begin{aligned}\implies 10^3+11^3 & =(10+11)(10^2-10 \cdot 11+11^2)\\& = 21(100-110+121)\\ & = 21(-10+121)\\ & = 21(111)\\& = 21 (100 + 10 + 1)\\ & = (21 \cdot 100)+(21 \cdot 10)+(21 \cdot 1)\\ & = 2100 +210+21\\ & = 2310 + 21\\ & = 2331\end{aligned}`