Problem: Evaluate each function at the given value using The Remainder Theorem.
The Remainder Theorem states that if a polynomial \( f(x) \) is divided by \( x - c \), the remainder of the division is equal to \( f(c) \). Therefore, to evaluate \( f(c) \), we simply substitute \( c \) into the polynomial \( f(x) \).
Let's solve each part step by step.
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####
Part (a):
Evaluate \( f(x) = x^3 + 2x^2 - 5x + 4 \) at \( x = -2 \).
1. Substitute \( x = -2 \) into the polynomial:
\[
f(-2) = (-2)^3 + 2(-2)^2 - 5(-2) + 4
\]
2. Calculate each term:
\[
(-2)^3 = -8, \quad 2(-2)^2 = 2(4) = 8, \quad -5(-2) = 10, \quad 4 = 4
\]
3. Add the terms together:
\[
f(-2) = -8 + 8 + 10 + 4 = 14
\]
Answer for Part (a):
\[
\boxed{14}
\]
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####
Part (b):
Evaluate \( f(x) = x^3 + 2x^2 - 5x + 4 \) at \( x = 2 \).
1. Substitute \( x = 2 \) into the polynomial:
\[
f(2) = (2)^3 + 2(2)^2 - 5(2) + 4
\]
2. Calculate each term:
\[
(2)^3 = 8, \quad 2(2)^2 = 2(4) = 8, \quad -5(2) = -10, \quad 4 = 4
\]
3. Add the terms together:
\[
f(2) = 8 + 8 - 10 + 4 = 10
\]
Answer for Part (b):
\[
\boxed{10}
\]
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####
Part (c):
Evaluate \( f(x) = x^3 - 6x^2 + 11x - 6 \) at \( x = 3 \).
1. Substitute \( x = 3 \) into the polynomial:
\[
f(3) = (3)^3 - 6(3)^2 + 11(3) - 6
\]
2. Calculate each term:
\[
(3)^3 = 27, \quad -6(3)^2 = -6(9) = -54, \quad 11(3) = 33, \quad -6 = -6
\]
3. Add the terms together:
\[
f(3) = 27 - 54 + 33 - 6 = 0
\]
Answer for Part (c):
\[
\boxed{0}
\]
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####
Part (d):
Evaluate \( f(x) = x^3 - 6x^2 + 11x - 6 \) at \( x = -1 \).
1. Substitute \( x = -1 \) into the polynomial:
\[
f(-1) = (-1)^3 - 6(-1)^2 + 11(-1) - 6
\]
2. Calculate each term:
\[
(-1)^3 = -1, \quad -6(-1)^2 = -6(1) = -6, \quad 11(-1) = -11, \quad -6 = -6
\]
3. Add the terms together:
\[
f(-1) = -1 - 6 - 11 - 6 = -24
\]
Answer for Part (d):
\[
\boxed{-24}
\]
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####
Part (e):
Evaluate \( f(x) = x^4 - 5x^3 + 2x^2 + 3x - 7 \) at \( x = 2 \).
1. Substitute \( x = 2 \) into the polynomial:
\[
f(2) = (2)^4 - 5(2)^3 + 2(2)^2 + 3(2) - 7
\]
2. Calculate each term:
\[
(2)^4 = 16, \quad -5(2)^3 = -5(8) = -40, \quad 2(2)^2 = 2(4) = 8, \quad 3(2) = 6, \quad -7 = -7
\]
3. Add the terms together:
\[
f(2) = 16 - 40 + 8 + 6 - 7 = -7
\]
Answer for Part (e):
\[
\boxed{-7}
\]
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####
Part (f):
Evaluate \( f(x) = x^4 - 5x^3 + 2x^2 + 3x - 7 \) at \( x = -2 \).
1. Substitute \( x = -2 \) into the polynomial:
\[
f(-2) = (-2)^4 - 5(-2)^3 + 2(-2)^2 + 3(-2) - 7
\]
2. Calculate each term:
\[
(-2)^4 = 16, \quad -5(-2)^3 = -5(-8) = 40, \quad 2(-2)^2 = 2(4) = 8, \quad 3(-2) = -6, \quad -7 = -7
\]
3. Add the terms together:
\[
f(-2) = 16 + 40 + 8 - 6 - 7 = 51
\]
Answer for Part (f):
\[
\boxed{51}
\]
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Final Answers:
\[
\boxed{14, 10, 0, -24, -7, 51}
\]
Parent Tip: Review the logic above to help your child master the concept of remainder theorem worksheet.
