Use the Remainder Theorem to find the value of f(12)
if f(x)=14x3−18x+3
.(1 point)
Responses
9532
Start Fraction 95 over 32 End Fraction
258
Start Fraction 25 over 8 End Fraction
194
Start Fraction 19 over 4 End Fraction
3
To use the Remainder Theorem, we substitute the value 12 into the given function:
f(12) = 14(12)^3 - 18(12) + 3
= 14(1728) - 216 + 3
= 24192 - 216 + 3
= 23879.
Therefore, the value of f(12) is 23879.
To find the value of f(12) using the Remainder Theorem, we need to divide the given polynomial f(x) by the factor corresponding to the desired value. In this case, we are dividing by (x - 12) since we want to find f(12).
Dividing 14x^3 - 18x + 3 by (x - 12) using polynomial long division:
14x^2 + 168x + 2016
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x - 12 | 14x^3 + 0x^2 - 18x + 3
-(14x^3 - 168x^2)
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168x^2 - 18x
-(168x^2 - 2016x)
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1998x + 3
-(1998x - 23976)
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23979
The remainder of this division is 23979, which represents f(12). Therefore, the value of f(12) is 23979.
To use the Remainder Theorem to find the value of f(12), we need to divide the given polynomial, f(x) = 14x^3 - 18x + 3, by the expression (x - 12).
Performing the division, we get:
(14x^3 - 18x + 3) ÷ (x - 12)
Using polynomial long division, we start by dividing the highest degree term:
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(x - 12) | 14x^3 - 18x + 3
-14x^2 is obtained by multiplying (x - 12) by -14x^2
-14x^2
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(x - 12) | 14x^3 - 18x + 3
Next, we subtract -14x^2(x - 12) from the original polynomial:
-14x^2
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(x - 12) | 14x^3 - 18x + 3
-(-14x^2) +0 + 0
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14x^3 + 14x^2 + 0x + 3
Now we bring down the next term, which is -18x:
-14x^2
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(x - 12) | 14x^3 - 18x + 3
-(-14x^2) +0 + 0
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14x^3 + 14x^2 + 0x + 3
- 14x^2 - 18x + 0
We can then repeat the process with the new polynomial:
-14x^2 - 2x
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(x - 12) | 14x^3 - 18x + 3
-(-14x^2) +0 + 0
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14x^3 + 14x^2 + 0x + 3
- 14x^2 - 18x + 0
- 14x^2 + 168x + 0
Finally, we bring down the last term, which is 3:
-14x^2 - 2x + 194
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(x - 12) | 14x^3 - 18x + 3
-(-14x^2) +0 + 0
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14x^3 + 14x^2 + 0x + 3
- 14x^2 - 18x + 0
- 14x^2 + 168x + 0
+ 3
At this point, we have reached a polynomial where the degree is less than the divisor (x - 12). Therefore, the remainder is 3.
The value of f(12) is equal to the remainder, which is 3.