Hi all,
What would the largest validity range possible for:
16x^2 - 128x^4 + 4096x^6/3! - 16384x^8 + 1048576x^10/5!
To find the largest validity range for an algebraic expression, we need to consider the values for which the expression is defined. In this case, the expression involves powers of x, so we need to determine the values of x for which these powers are valid.
To do this, we can look at the exponent of x in each term of the expression. The exponent of x determines the type of polynomial, and it helps us determine the possible values of x that make the expression valid.
Let's analyze each term one by one:
1. 16x^2:
The exponent of x is 2, so x can take any real value. There is no restriction on x for this term.
2. -128x^4:
The exponent of x is 4, so x can take any real value. There is no restriction on x for this term.
3. 4096x^6/3!:
The exponent of x is 6. The factor 3! (or 3 factorial) in the denominator means we divide by 3 × 2 × 1 (or 6). This indicates that x can take any real value.
4. -16384x^8:
The exponent of x is 8, so x can take any real value. There is no restriction on x for this term.
5. 1048576x^10/5!:
The exponent of x is 10. The factor 5! (or 5 factorial) in the denominator means we divide by 5 × 4 × 3 × 2 × 1 (or 120). This indicates that x can take any real value.
From the analysis above, we can see that there are no restrictions on x for each term of the expression. Therefore, the largest validity range for the given expression is the set of all real numbers (-∞, +∞).