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using the polynomiak identity (x^2+y^2)^2=(x^2-y^2+(2xy)^2 to generate a pythagereorm triple where x equals 10 and y equals 6 which of the following is one of the values of the pythagereom theorem

A.120
B.16
C.256
D.60

3 answers

To generate a Pythagorean triple using the polynomial identity \((x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2\), we will substitute \(x = 10\) and \(y = 6\).

  1. First, calculate \(x^2\) and \(y^2\): \[ x^2 = 10^2 = 100 \] \[ y^2 = 6^2 = 36 \]

  2. Now, calculate \(x^2 + y^2\): \[ x^2 + y^2 = 100 + 36 = 136 \]

  3. Next, calculate \(x^2 - y^2\): \[ x^2 - y^2 = 100 - 36 = 64 \]

  4. Then calculate \(2xy\): \[ 2xy = 2 \cdot 10 \cdot 6 = 120 \]

Now we have:

  • \( (x^2 - y^2) = 64 \)
  • \( (2xy) = 120 \)

According to the identity: \[ (x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2 \] We can confirm that: \[ 136^2 = 64^2 + 120^2 \]

Calculating both sides:

  • Left side: \[ 136^2 = 18496 \]

  • Right side: \[ 64^2 = 4096 \quad \text{and} \quad 120^2 = 14400 \] \[ 64^2 + 120^2 = 4096 + 14400 = 18496 \]

Both sides match, confirming that the identity holds.

The values of the Pythagorean triple generated are: \[ a = x^2 - y^2 = 64, \quad b = 2xy = 120, \quad c = x^2 + y^2 = 136 \]

Now, checking the options:

  • A. 120 (one of the values)
  • B. 16
  • C. 256
  • D. 60

Thus, the answer is A. 120.

Are you sure

Yes, let's go through it again step by step to ensure the calculations are correct and clarify the result:

We use the polynomial identity: \[ (x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2 \] with \(x = 10\) and \(y = 6\).

  1. Calculate \(x^2\) and \(y^2\): \[ x^2 = 10^2 = 100 \] \[ y^2 = 6^2 = 36 \]

  2. Calculate \(x^2 + y^2\): \[ x^2 + y^2 = 100 + 36 = 136 \]

  3. Calculate \(x^2 - y^2\): \[ x^2 - y^2 = 100 - 36 = 64 \]

  4. Calculate \(2xy\): \[ 2xy = 2 \cdot 10 \cdot 6 = 120 \]

So, we have:

  • \(x^2 + y^2 = 136\)
  • \(x^2 - y^2 = 64\)
  • \(2xy = 120\)

Now, we can confirm the Pythagorean triple:

  • The expression states that \( (x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2 \).
  • We already calculated:
    • Left side: \[ (136)^2 = 18496 \]
    • Right side: \[ (64)^2 + (120)^2 = 4096 + 14400 = 18496 \]

So, both sides equal \(18496\), confirming this identity is valid.

The values of the Pythagorean triple are:

  • \(a = 64\)
  • \(b = 120\)
  • \(c = 136\)

Looking at your original options:

  • A. 120 (this is one of the values)
  • B. 16
  • C. 256
  • D. 60

Thus, indeed, the answer is A. 120. It represents one of the numeric components of the generated Pythagorean triple. If there is any specific part you want clarification on, please let me know!