Question

Task 2

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

Part 1. Let's say we choose the two-digit number 78. We can rewrite 78 as the difference of two numbers by breaking it down into two-digit numbers that sum up to 78. For example, we can express 78 as 80-2.

Hope this help

Using the identity (x - y)2 = x2 - 2xy + y2, we can square 78 without using a calculator. Applying the formula, we have:
(80 - 2)2 = 80^2 - 2 * 80 * 2 + 2^2

Simplifying this expression, we get:
80^2 - 2 * 80 * 2 + 2^2 = 6400 - 320 + 4 = 6084

Therefore, squaring the number 78 using the given identity yields the result 6084.

Part 2. Let's choose two values, a = 10 and b = 12. The sum of the cubes of these numbers can be calculated by using the identity a3 + b3 = (a + b)(a2 - ab + b2).

Applying the formula, we have:
(10 + 12)(10^2 - 10 * 12 + 12^2)

Simplifying this expression, we get:
(22)(100 - 120 + 144) = 22 * 124 = 2728

Therefore, the sum of the cubes of the numbers 10 and 12, using the given identity, gives us the result 2728.