Answer :

Sure! Let's find [tex]\( f(2) \)[/tex] for the polynomial [tex]\( f(x) = 3x^6 - 2x^3 + 6x \)[/tex] using the Remainder Theorem.

The Remainder Theorem states that if you have a polynomial [tex]\( f(x) \)[/tex] and you divide it by [tex]\( x - c \)[/tex], the remainder of this division is [tex]\( f(c) \)[/tex].

To find [tex]\( f(2) \)[/tex], we will substitute [tex]\( x = 2 \)[/tex] into the polynomial:

[tex]\[ f(x) = 3x^6 - 2x^3 + 6x \][/tex]

Substitute [tex]\( x = 2 \)[/tex]:

[tex]\[ f(2) = 3(2)^6 - 2(2)^3 + 6(2) \][/tex]

Let's calculate each term step-by-step:

1. Calculate [tex]\( (2)^6 \)[/tex]:
[tex]\[ 2^6 = 64 \][/tex]

So,
[tex]\[ 3(2)^6 = 3 \times 64 = 192 \][/tex]

2. Calculate [tex]\( (2)^3 \)[/tex]:
[tex]\[ 2^3 = 8 \][/tex]

So,
[tex]\[ -2(2)^3 = -2 \times 8 = -16 \][/tex]

3. Calculate [tex]\( 6(2) \)[/tex]:
[tex]\[ 6 \times 2 = 12 \][/tex]

Now, add up these results:

[tex]\[ f(2) = 192 - 16 + 12 \][/tex]

Combine the numbers:
[tex]\[ 192 - 16 = 176 \][/tex]
[tex]\[ 176 + 12 = 188 \][/tex]

So, [tex]\( f(2) = 188 \)[/tex].

Therefore, the value of [tex]\( f(2) \)[/tex] is [tex]\(\boxed{188}\)[/tex].

Thanks for taking the time to read Compute using the Remainder Theorem Find tex f 2 tex for tex f x 3x 6 2x 3 6x tex A 180 B 188 C. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada