If f, left bracket, x, right bracket, equals, x, cubed, plus, 10, x, squared, plus, 23, x, plus, 14
f(x)=x^3+10x^2+23x+14, which of the following is not a factor of f, left bracket, x, right bracketf(x)?
Answer
Multiple Choice Answers
left bracket, x, plus, 7, right bracket(x+7)
left bracket, x, plus, 1, right bracket(x+1)
left bracket, x, minus, 2, right bracket(x−2)
left bracket, x, plus, 2, right bracket(x+2)
To find which of the given options is not a factor of f(x), we can try plugging in each option into f(x) and see if it equals to 0.
Let's substitute x+7 into f(x):
f(x) = (x+7)^3 + 10(x+7)^2 + 23(x+7) + 14
Expanding this expression and simplifying, we get:
f(x) = x^3 + 21x^2 + 133x + 280
Since this is not equal to 0, x+7 is not a factor of f(x).
Now, let's substitute x+1 into f(x):
f(x) = (x+1)^3 + 10(x+1)^2 + 23(x+1) + 14
Expanding this expression and simplifying, we get:
f(x) = x^3 + 13x^2 + 44x + 48
Since this is not equal to 0, x+1 is not a factor of f(x).
Now, let's substitute x-2 into f(x):
f(x) = (x-2)^3 + 10(x-2)^2 + 23(x-2) + 14
Expanding this expression and simplifying, we get:
f(x) = x^3 + 4x^2 - 13x - 12
Since this is not equal to 0, x-2 is not a factor of f(x).
Finally, let's substitute x+2 into f(x):
f(x) = (x+2)^3 + 10(x+2)^2 + 23(x+2) + 14
Expanding this expression and simplifying, we get:
f(x) = x^3 + 6x^2 + 5x + 8
Since this is not equal to 0, x+2 is not a factor of f(x).
Therefore, the option that is not a factor of f(x) is:
[x+7] (x+7)
To find out which of the given expressions is not a factor of f(x), we can use the Remainder Theorem:
1. Take the expression inside each pair of brackets and set it equal to zero to solve for x:
a) x + 7 = 0, solve for x: x = -7
b) x + 1 = 0, solve for x: x = -1
c) x - 2 = 0, solve for x: x = 2
d) x + 2 = 0, solve for x: x = -2
2. Evaluate f(x) for each value of x obtained in Step 1:
a) f(-7) = (-7)^3 + 10(-7)^2 + 23(-7) + 14
b) f(-1) = (-1)^3 + 10(-1)^2 + 23(-1) + 14
c) f(2) = (2)^3 + 10(2)^2 + 23(2) + 14
d) f(-2) = (-2)^3 + 10(-2)^2 + 23(-2) + 14
3. If the value obtained is zero for any of the expressions in Step 2, it means that expression is a factor of f(x). If the value is non-zero, then it is not a factor.
Now, we can evaluate f(x) for each of the values:
a) f(-7) = -21 + 490 - 161 + 14 = 322 (non-zero)
b) f(-1) = -1 + 10 + 23 - 14 = 18 (non-zero)
c) f(2) = 8 + 40 + 46 + 14 = 108 (non-zero)
d) f(-2) = -8 + 40 - 46 + 14 = 0 (zero)
Based on the results, the expression (x + 2) is a factor of f(x) since f(-2) gives a remainder of zero. Therefore, the correct answer is:
left bracket, x, plus, 2, right bracket (x + 2)
To determine which of the given expressions is not a factor of f(x), we can use the remainder theorem. According to the remainder theorem, if (x-a) is a factor of a polynomial f(x), then f(a) will be equal to zero.
Let's substitute the values of the given options in the polynomial f(x) and check which one does not result in zero.
1. Substitute (x+7) into f(x):
f(x) = (x+7)^3 + 10(x+7)^2 + 23(x+7) + 14
2. Substitute (x+1) into f(x):
f(x) = (x+1)^3 + 10(x+1)^2 + 23(x+1) + 14
3. Substitute (x-2) into f(x):
f(x) = (x-2)^3 + 10(x-2)^2 + 23(x-2) + 14
4. Substitute (x+2) into f(x):
f(x) = (x+2)^3 + 10(x+2)^2 + 23(x+2) + 14
We need to check if any of these substitutions result in zero.
After substituting and simplifying, we find that option 1, (x+7), results in zero, which means (x+7) is a factor of f(x).
Option 2, (x+1), also results in zero after substituting.
Option 4, (x+2), also results in zero.
However, option 3, (x-2), does not result in zero after substituting. Thus, we conclude that (x-2) is not a factor of f(x).
Therefore, the answer is option left bracket, x, minus, 2, right bracket (x−2).