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Answer :
To find [tex]\(f(-3)\)[/tex] and [tex]\(f(4)\)[/tex] for the function [tex]\(f(x) = x^3 + 3x^2 + 2x - 50\)[/tex] using synthetic substitution, follow these steps:
1. Evaluate [tex]\(f(-3)\)[/tex]:
- Start with [tex]\(f(x) = x^3 + 3x^2 + 2x - 50\)[/tex].
- Substitute [tex]\(x = -3\)[/tex] into the polynomial:
[tex]\[
f(-3) = (-3)^3 + 3(-3)^2 + 2(-3) - 50
\][/tex]
- Calculate each term:
- [tex]\((-3)^3 = -27\)[/tex]
- [tex]\(3(-3)^2 = 3 \times 9 = 27\)[/tex]
- [tex]\(2(-3) = -6\)[/tex]
- Combine all terms: [tex]\(-27 + 27 - 6 - 50\)[/tex]
- Simplify the expression:
[tex]\(-27 + 27 = 0\)[/tex]
[tex]\(0 - 6 = -6\)[/tex]
[tex]\(-6 - 50 = -56\)[/tex]
- Therefore, [tex]\(f(-3) = -56\)[/tex].
2. Evaluate [tex]\(f(4)\)[/tex]:
- Again, use the function [tex]\(f(x) = x^3 + 3x^2 + 2x - 50\)[/tex].
- Substitute [tex]\(x = 4\)[/tex]:
[tex]\[
f(4) = (4)^3 + 3(4)^2 + 2(4) - 50
\][/tex]
- Calculate each term:
- [tex]\((4)^3 = 64\)[/tex]
- [tex]\(3(4)^2 = 3 \times 16 = 48\)[/tex]
- [tex]\(2(4) = 8\)[/tex]
- Combine all terms: [tex]\(64 + 48 + 8 - 50\)[/tex]
- Simplify the expression:
[tex]\(64 + 48 = 112\)[/tex]
[tex]\(112 + 8 = 120\)[/tex]
[tex]\(120 - 50 = 70\)[/tex]
- Therefore, [tex]\(f(4) = 70\)[/tex].
In summary, the values are [tex]\(f(-3) = -56\)[/tex] and [tex]\(f(4) = 70\)[/tex]. These results match one of the given options, [tex]\(-56, 70\)[/tex].
1. Evaluate [tex]\(f(-3)\)[/tex]:
- Start with [tex]\(f(x) = x^3 + 3x^2 + 2x - 50\)[/tex].
- Substitute [tex]\(x = -3\)[/tex] into the polynomial:
[tex]\[
f(-3) = (-3)^3 + 3(-3)^2 + 2(-3) - 50
\][/tex]
- Calculate each term:
- [tex]\((-3)^3 = -27\)[/tex]
- [tex]\(3(-3)^2 = 3 \times 9 = 27\)[/tex]
- [tex]\(2(-3) = -6\)[/tex]
- Combine all terms: [tex]\(-27 + 27 - 6 - 50\)[/tex]
- Simplify the expression:
[tex]\(-27 + 27 = 0\)[/tex]
[tex]\(0 - 6 = -6\)[/tex]
[tex]\(-6 - 50 = -56\)[/tex]
- Therefore, [tex]\(f(-3) = -56\)[/tex].
2. Evaluate [tex]\(f(4)\)[/tex]:
- Again, use the function [tex]\(f(x) = x^3 + 3x^2 + 2x - 50\)[/tex].
- Substitute [tex]\(x = 4\)[/tex]:
[tex]\[
f(4) = (4)^3 + 3(4)^2 + 2(4) - 50
\][/tex]
- Calculate each term:
- [tex]\((4)^3 = 64\)[/tex]
- [tex]\(3(4)^2 = 3 \times 16 = 48\)[/tex]
- [tex]\(2(4) = 8\)[/tex]
- Combine all terms: [tex]\(64 + 48 + 8 - 50\)[/tex]
- Simplify the expression:
[tex]\(64 + 48 = 112\)[/tex]
[tex]\(112 + 8 = 120\)[/tex]
[tex]\(120 - 50 = 70\)[/tex]
- Therefore, [tex]\(f(4) = 70\)[/tex].
In summary, the values are [tex]\(f(-3) = -56\)[/tex] and [tex]\(f(4) = 70\)[/tex]. These results match one of the given options, [tex]\(-56, 70\)[/tex].
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