Answer :
To find [tex]\( F(2) \)[/tex] for the polynomial function [tex]\( F(x) = 3x^3 - 5x^2 + 13x - 19 \)[/tex] using the Remainder Theorem, follow these steps:
1. Understand the Remainder Theorem: It states that if you divide a polynomial [tex]\( F(x) \)[/tex] by [tex]\( x - c \)[/tex], then the remainder is [tex]\( F(c) \)[/tex]. In other words, to find the value of the polynomial at [tex]\( x = c \)[/tex], simply substitute [tex]\( c \)[/tex] into the polynomial.
2. Substitute 2 into the Polynomial Function: We want to find [tex]\( F(2) \)[/tex], so substitute [tex]\( x = 2 \)[/tex] into the polynomial:
[tex]\[
F(2) = 3(2)^3 - 5(2)^2 + 13(2) - 19
\][/tex]
3. Calculate Each Term:
- First term: [tex]\( 3(2)^3 = 3 \times 8 = 24 \)[/tex]
- Second term: [tex]\( -5(2)^2 = -5 \times 4 = -20 \)[/tex]
- Third term: [tex]\( 13(2) = 26 \)[/tex]
4. Combine the Terms:
[tex]\[
F(2) = 24 - 20 + 26 - 19
\][/tex]
Simplify step-by-step:
- [tex]\( 24 - 20 = 4 \)[/tex]
- [tex]\( 4 + 26 = 30 \)[/tex]
- [tex]\( 30 - 19 = 11 \)[/tex]
5. Conclusion: The value of [tex]\( F(2) \)[/tex] is 11.
Thus, the final answer is 11, which corresponds to option A.
1. Understand the Remainder Theorem: It states that if you divide a polynomial [tex]\( F(x) \)[/tex] by [tex]\( x - c \)[/tex], then the remainder is [tex]\( F(c) \)[/tex]. In other words, to find the value of the polynomial at [tex]\( x = c \)[/tex], simply substitute [tex]\( c \)[/tex] into the polynomial.
2. Substitute 2 into the Polynomial Function: We want to find [tex]\( F(2) \)[/tex], so substitute [tex]\( x = 2 \)[/tex] into the polynomial:
[tex]\[
F(2) = 3(2)^3 - 5(2)^2 + 13(2) - 19
\][/tex]
3. Calculate Each Term:
- First term: [tex]\( 3(2)^3 = 3 \times 8 = 24 \)[/tex]
- Second term: [tex]\( -5(2)^2 = -5 \times 4 = -20 \)[/tex]
- Third term: [tex]\( 13(2) = 26 \)[/tex]
4. Combine the Terms:
[tex]\[
F(2) = 24 - 20 + 26 - 19
\][/tex]
Simplify step-by-step:
- [tex]\( 24 - 20 = 4 \)[/tex]
- [tex]\( 4 + 26 = 30 \)[/tex]
- [tex]\( 30 - 19 = 11 \)[/tex]
5. Conclusion: The value of [tex]\( F(2) \)[/tex] is 11.
Thus, the final answer is 11, which corresponds to option A.
