To find the remainder when dividing the polynomial [tex]\(5x^2 + 9x - 2\)[/tex] by [tex]\(x + 3\)[/tex], we use the Remainder Theorem. The Remainder Theorem states that the remainder of the division of a polynomial [tex]\( f(x) \)[/tex] by [tex]\( x - a \)[/tex] is given by [tex]\( f(a) \)[/tex].
Here, the divisor is [tex]\( x + 3 \)[/tex], which can be written as [tex]\( x - (-3) \)[/tex]. Therefore, [tex]\( a = -3 \)[/tex].
Now, we need to evaluate the polynomial [tex]\( 5x^2 + 9x - 2 \)[/tex] at [tex]\( x = -3 \)[/tex].
1. Plug [tex]\( x = -3 \)[/tex] into the polynomial:
[tex]\[
f(x) = 5x^2 + 9x - 2
\][/tex]
2. Substitute [tex]\( x = -3 \)[/tex]:
[tex]\[
f(-3) = 5(-3)^2 + 9(-3) - 2
\][/tex]
3. Calculate each term:
[tex]\[
5(-3)^2 = 5 \times 9 = 45
\][/tex]
[tex]\[
9(-3) = -27
\][/tex]
[tex]\[
-2 = -2
\][/tex]
4. Add the results together:
[tex]\[
f(-3) = 45 - 27 - 2 = 16
\][/tex]
Thus, the remainder when [tex]\( 5x^2 + 9x - 2 \)[/tex] is divided by [tex]\( x + 3 \)[/tex] is [tex]\( 16 \)[/tex].
Hence, the correct answer is:
B. 16
