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Which of the following polynomials can approximate the expression (4x3+5)(3x6−8x2)2x2+4x3+13 using first and last terms only?(1 point) Responses 4x3+...13 6x7+...−7 6 x superscript 7 baseline plus 13 12x9+...13 12 x superscript 7 baseline minus 27 6x7+...4x3

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To approximate the expression \( (4x^3 + 5)(3x^6 - 8x^2)2x^2 + 4x^3 + 13 \) using the first and last terms only, we will focus on the leading (highest degree) term and the constant (lowest degree) term of each part of the expression.

Step 1: Find the first and last terms of each polynomial component

  1. First Component: \( (4x^3 + 5) \)

    • First term: \( 4x^3 \)
    • Last term (constant): \( 5 \)

  2. Second Component: \( (3x^6 - 8x^2) \)

    • First term: \( 3x^6 \)
    • Last term: \( -8x^2 \)

  3. Third Component: \( 2x^2 \)

    • First term: \( 2x^2 \)
    • Last term: \( 2x^2 \) (it's just \( 2x^2 \))

  4. Fourth Component: \( 4x^3 \)

    • First term: \( 4x^3 \)
    • Last term: \( 4x^3 \)

  5. Fifth Component: \( 13 \)

    • First term: \( 13 \)
    • Last term: \( 13 \) (constant)

Step 2: Multiply the leading terms

Next, we compute the product of the leading terms:

  • Leading term of \( (4x^3)(3x^6)(2x^2) \): \[ 4x^3 \cdot 3x^6 \cdot 2x^2 = 24x^{11} \]

Step 3: Add the last terms

Now, we consider the constant (last) terms:

  • Last term: \( 5 \) (from the first component) and \( 13 \) (from the constant component): \[ 5 + 13 = 18 \]

Step 4: Determine the approximate polynomial

The resulting approximation of the expression, using just the first and last terms, would be dominated by the leading term and added to the constant.

Thus, the approximate polynomial can be expressed as: \[ 24x^{11} + 18 \]

However, it's important to check which of the provided options comes closest to this without necessarily being accurate.

Review Provided Responses

  1. \( 4x^3 + ... + 13 \)
  2. \( 6x^7 + ... - 7 \)
  3. \( 6x^7 + ... + 13 \)
  4. \( 12x^9 + ... - 27 \)
  5. \( 12x^7 + ... - 13 \)
  6. \( 6x^7 + ... + 4x^3 \)

Conclusion

There is no exact match for \( 24x^{11} + 18 \). The expression seems to provide options with a leading term of \( x^7 \) or \( x^9 \) which are lower in degree compared with \( 24x^{11} \).

Since none of the options match the resulting expression, we should ideally conclude that none of the above options offer a close approximation based on the leading terms calculated.

If an approximation were needed and based on the closeness of leading terms, \( 6x^7 + ... + 4x^3 \) is the best reasonable response due to its leading term structure, but it might not remain accurate to the calculation done.

If you have to select from above, choose the one that refers to \( x^7 \).