Answer :

To solve the problem, we need to evaluate the polynomial function [tex]\( g(x) = 8x^5 - 58x^4 + 60x^3 + 140 \)[/tex] at [tex]\( x = 6 \)[/tex].

Here are the steps to find [tex]\( g(6) \)[/tex]:

1. Substitute [tex]\( x \)[/tex] with 6 in the polynomial:

[tex]\[
g(6) = 8(6)^5 - 58(6)^4 + 60(6)^3 + 140
\][/tex]

2. Calculate each term:

- [tex]\( 6^5 = 7776 \)[/tex]
- Multiply by 8: [tex]\( 8 \times 7776 = 62208 \)[/tex]

- [tex]\( 6^4 = 1296 \)[/tex]
- Multiply by 58: [tex]\( 58 \times 1296 = 75168 \)[/tex]

- [tex]\( 6^3 = 216 \)[/tex]
- Multiply by 60: [tex]\( 60 \times 216 = 12960 \)[/tex]

3. Combine these results:

Now put these values into the expression:
[tex]\[
g(6) = 62208 - 75168 + 12960 + 140
\][/tex]

4. Perform the addition and subtraction:

- Start with [tex]\( 62208 - 75168 = -12960 \)[/tex]
- Then add [tex]\( 12960 \)[/tex]: [tex]\(-12960 + 12960 = 0\)[/tex]
- Finally, add 140: [tex]\( 0 + 140 = 140 \)[/tex]

Hence, the value of [tex]\( g(6) \)[/tex] is [tex]\( 140 \)[/tex].

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