Explain why the product of an integer multiplied by itself can never be negative. Show your thinking by using words and numbers.
When we multiply two numbers, the result is positive if both numbers have the same sign (either positive or negative), and the result is negative if the numbers have opposite signs (one positive and one negative).
Now, let's consider the product of an integer multiplied by itself. We can write this as (n)(n), where n is an integer.
Case 1: n is positive
If n is positive, then both factors in the product (n and n) have the same sign, so the result will be positive. For example, if we multiply 3 by itself, we get 3 * 3 = 9, which is positive.
Case 2: n is negative
If n is negative, then both factors in the product (n and n) have the same sign, so the result will be positive. For example, if we multiply -2 by itself, we get (-2) * (-2) = 4, which is positive.
In both cases, whether n is positive or negative, the product of n multiplied by itself is always positive. Therefore, the product of an integer multiplied by itself can never be negative.