Question

Decide whether each of the following pairs of expressions are equivalent for all values of x (or a and b). If they are equivalent, show how you can be sure. If they are not, justify your reasoning completely.

a) (x³)² and x² - 9
b) (x⁴)² and x² - 8x + 16
c) (2 - 1)(2x - 3) and 2x² - 3
d) 3(x - 4)² - 2 and 3x² - 24x + 50

Answer 1

After evaluating each pair of expressions, it is clear that none of the pairs are equivalent for all values of x due to different algebraic forms and results.

Step-by-step explanation:

Let's evaluate each pair of expressions to see if they are equivalent for all values of x (or a and b).

1. (x³)² vs. x² - 9: The first expression simplifies to x¶ (since raising a power to a power multiplies the exponents), which is not equivalent to x² - 9 because they are fundamentally different algebraic forms and will yield different results for most values of x.
2. (x⁴)² vs. x² - 8x + 16: Similar to the first case, the left expression simplifies to x⁸, which is not equivalent to x² - 8x + 16, the latter being the expansion of (x - 4)², and thus different for most values of x.
3. (2 - 1)(2x - 3) vs. 2x² - 3: Simplifying the left expression gives us 2x - 3 which is clearly not equivalent to 2x² - 3 as one is linear and the other is quadratic.
4. 3(x - 4)² - 2 vs. 3x² - 24x + 50: Expanding the first expression gives 3(x² - 8x + 16) - 2, which simplifies to 3x² - 24x + 48 - 2, equaling 3x² - 24x + 46, not 3x² - 24x + 50, hence they are not equivalent.

In conclusion, none of the pairs of expressions provided are equivalent.