Question

mr. factorstein teaches an algebra class. he wants to show a numerical example that illustrates the identity $a^2 - b^2 = (a - b)(a + b)$. which of the following would be the best example to use?
a. divide 500 by 100.
b. subtract 30 from 300.
c. add 250 to 350.
d. multiply 103 by 97.
(d) let $a = 100$ and let $b = 3$. then $a - b = 97$ and $a + b = 103$. then the product of 103 and 97 can be calculated as $100^2 - 3^2 = 10,000 - 9 = 9991$.

Explanation:

The identity given is \(a^{2}-b^{2}=(a - b)(a + b)\). We need to check which option can be expressed in this form.

Step 1: Analyze Option A

For option A: Divide 500 by 100. This is a division operation, \(500\div100 = 5\), which has nothing to do with the difference of squares identity.

Step 2: Analyze Option B

For option B: Subtract 30 from 300, \(300 - 30=270\), this is a simple subtraction and not related to the identity \(a^{2}-b^{2}=(a - b)(a + b)\).

Step 3: Analyze Option C

For option C: Add 250 to 350, \(250+350 = 600\), this is an addition operation and not related to the difference of squares identity.

Step 4: Analyze Option D

For option D: Multiply 103 by 97. We can write \(103=100 + 3\) and \(97=100-3\). So, \(103\times97=(100 + 3)(100 - 3)\). By the identity \(a^{2}-b^{2}=(a - b)(a + b)\), if \(a = 100\) and \(b = 3\), then \((100+3)(100 - 3)=100^{2}-3^{2}=10000 - 9=9991\), which is the product of 103 and 97. So this operation can be represented using the difference of squares identity.

Answer:

D. Multiply 103 by 97.