Which of the following polynomials can approximate the expression (4x3+5)(3x6−8x2)2x2+4x3+13

Question

(
4
𝑥
3
+
5
)
(
3
𝑥
6
−
8
𝑥
2
)
2
𝑥
2
+
4
𝑥
3
+
13
using first and last terms only?(1 point)
Responses

6x7+…−7
6
𝑥
7
+
…
−
7
6 x to the 7th power plus dot dot dot minus 7

6x7+…4x3
6
𝑥
7
+
…
4
𝑥
3
6 x to the 7th power plus dot dot dot 4 x cubed

4x3+…13
4
𝑥
3
+
…
13
4 x cubed plus dot dot dot 13

12x9+…13

Answer 1

To approximate the expression \((4x^3 + 5)(3x^6 - 8x^2) + 2x^2 + 4x^3 + 13\) using only the first and last terms, we first identify the highest degree term (the first term) and the constant term (the last term).

1. First term: The leading term from the product \((4x^3)(3x^6)\): \[ 4x^3 \cdot 3x^6 = 12x^9 \]

2. Last term: The constant term from the expression, which is \(13\).

Now we seek a polynomial that includes \(12x^9\) as the first term and \(13\) as the last term.

Among the response options:

1. \(6x^7 + … - 7\) does not have \(12x^9\) or \(13\).
2. \(6x^7 + … + 4x^3\) does not have \(12x^9\) or \(13\).
3. \(4x^3 + … + 13\) does not have \(12x^9\) as the first term.
4. \(12x^9 + … + 13\) does have \(12x^9\) as the first term and \(13\) as the last term.

Therefore, the polynomial that can approximate the expression using the first and last terms is:

12x^9 + ... + 13