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Algebraic identities are used both for transforming products of polynomials and for factorising polynomials into simpler factors. These formulas can be written as follows:
Solution.
The first term 16x² can be written as (4x)².
The second term 25y² can be written as (5y)².
Using the difference of squares formula a² − b² = (a − b)(a + b), we get:
16x² − 25y² = (4x)² − (5y)² = (4x − 5y)(4x + 5y)
Solution.
Notice that 4x² = (2x)² and 9y² = (3y)².
To use the square of a sum formula a² + 2ab + b² = (a + b)², check whether 12xy equals 2·(2x)·(3y).
Indeed: 12xy = 2 · (2x)(3y).
So, we can write:
4x² + 12xy + 9y² = (2x)² + 2·(2x)(3y) + (3y)² = (2x + 3y)²
Thus, the polynomial is a perfect square.
1. What is the value of the product:
(1 − 1/4)(1 − 1/9)(1 − 1/16)…(1 − 1/225)?
2. Two different numbers x and y (not necessarily integers) satisfy:
x² − 2000x = y² − 2000y.
Find the sum x + y.
3. Find all pairs of prime numbers such that the difference of their squares is also a prime number.
4. In how many ways can the number 23 be expressed as the difference of two squares of natural numbers?
5. Suppose a, b, c are three numbers such that:
a + b + c = 0.
Prove that in this case:
ab + ac + bc ≤ 0.
6. Factorise the expression:
x⁴ + 16.
7. It is known that:
a + b + c = 5, and ab + bc + ac = 5.
What can the value of a² + b² + c² be?
8. Prove that the product of four consecutive natural numbers, increased by 1, is always a perfect square.
