To determine if the given number [tex]\( c \)[/tex] is a zero of the polynomial [tex]\( f(x) = 2x^4 + 7x^3 - 18x^2 + 4x + 95 \)[/tex], we will employ the Remainder Theorem. The Remainder Theorem states that if a polynomial [tex]\( f(x) \)[/tex] is divided by [tex]\( x - c \)[/tex], the remainder of this division is [tex]\( f(c) \)[/tex]. If [tex]\( f(c) = 0 \)[/tex], then [tex]\( c \)[/tex] is a zero of the polynomial.
### Part 1: [tex]\( c = -5 \)[/tex]
1. Evaluate the polynomial at [tex]\( c = -5 \)[/tex]:
[tex]\[
f(-5) = 2(-5)^4 + 7(-5)^3 - 18(-5)^2 + 4(-5) + 95
\][/tex]
2. Calculate each term individually:
[tex]\[
2(-5)^4 = 2 \cdot 625 = 1250
\][/tex]
[tex]\[
7(-5)^3 = 7 \cdot (-125) = -875
\][/tex]
[tex]\[
-18(-5)^2 = -18 \cdot 25 = -450
\][/tex]
[tex]\[
4(-5) = -20
\][/tex]
[tex]\[
95 \text{ (constant term)} = 95
\][/tex]
3. Sum these values:
[tex]\[
f(-5) = 1250 - 875 - 450 - 20 + 95 = 0
\][/tex]
Since [tex]\( f(-5) = 0 \)[/tex], [tex]\( c = -5 \)[/tex] is a zero of the polynomial.
(a) [tex]\( c = -5 \)[/tex] is a zero of the polynomial.
### Part 2: [tex]\( c = 6 \)[/tex]
1. Evaluate the polynomial at [tex]\( c = 6 \)[/tex]:
[tex]\[
f(6) = 2(6)^4 + 7(6)^3 - 18(6)^2 + 4(6) + 95
\][/tex]
2. Calculate each term individually:
[tex]\[
2(6)^4 = 2 \cdot 1296 = 2592
\][/tex]
[tex]\[
7(6)^3 = 7 \cdot 216 = 1512
\][/tex]
[tex]\[
-18(6)^2 = -18 \cdot 36 = -648
\][/tex]
[tex]\[
4(6) = 24
\][/tex]
[tex]\[
95 \text{ (constant term)} = 95
\][/tex]
3. Sum these values:
[tex]\[
f(6) = 2592 + 1512 - 648 + 24 + 95 = 3575
\][/tex]
Since [tex]\( f(6) \neq 0 \)[/tex], [tex]\( c = 6 \)[/tex] is not a zero of the polynomial.
(b) [tex]\( c = 6 \)[/tex] is not a zero of the polynomial.
