Questions: Simplify. Use a graphing calculator table to verify your work. (x-11)^2 (x-11)^2=

Solution Steps

To simplify \((x-11)^2\), we can expand the expression using the binomial theorem or by multiplying the binomial by itself. Then, we can use Python to verify the result by evaluating the expression for different values of \(x\).

1. Expand the binomial \((x-11)^2\).
2. Verify the expanded form by evaluating it for different values of \(x\) using Python.

We start with the expression \((x - 11)^2\). Using the binomial expansion, we have:

\[ (x - 11)^2 = x^2 - 2 \cdot 11 \cdot x + 11^2 = x^2 - 22x + 121 \]

To verify the correctness of the expansion, we evaluate both the original expression \((x - 11)^2\) and the expanded form \(x^2 - 22x + 121\) for various values of \(x\):

• For \(x = 0\):

  • Original: \((0 - 11)^2 = 121\)
  • Expanded: \(0^2 - 22 \cdot 0 + 121 = 121\)

• For \(x = 1\):

  • Original: \((1 - 11)^2 = 100\)
  • Expanded: \(1^2 - 22 \cdot 1 + 121 = 100\)

• For \(x = 2\):

  • Original: \((2 - 11)^2 = 81\)
  • Expanded: \(2^2 - 22 \cdot 2 + 121 = 81\)

• For \(x = 10\):

  • Original: \((10 - 11)^2 = 1\)
  • Expanded: \(10^2 - 22 \cdot 10 + 121 = 1\)

• For \(x = 11\):

  • Original: \((11 - 11)^2 = 0\)
  • Expanded: \(11^2 - 22 \cdot 11 + 121 = 0\)

• For \(x = 12\):

  • Original: \((12 - 11)^2 = 1\)
  • Expanded: \(12^2 - 22 \cdot 12 + 121 = 1\)

In all cases, the original and expanded expressions yield the same results, confirming the correctness of the expansion.

Final Answer