Find the value of a, if (x+a) is a factor of x^3+ax^2-2x+a+4.
Answers
Answer:
If x+a is a factor of x³+ax²-2x+a+4 then a equals?
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x^3+ax^2-2x+a+4 Must be divisible by x+a
so,( x3+ax2−2x+a+4)/(x+a)
=x^2–2 is quotient and (a+4)-(-2a) is remainder
But the polynomial is divisible so the remainder should be 0
Thus , (a+4)-(-2a)=0
=>(a+4+2a)=0
=>3a+4=0
=>a=-4/3
The polynomial division done in this answer can be easily under stood by the following video:
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That is a very easy question.
Theorem
To solve this question, we would need to apply the Factor Theorem.
If you don't know the Factor Theorem:
Factor Theorem
It states that if f(x) is a polynomial and g(x) is the factor of f(x), where the degree of g(x) must be smaller than f(x), then the value/values of x when g(x) is equated to zero is/are the zeroes of the polynomial.
As f(x)=g(x)*q(x) + 0 ,[q(x) is the other factor/factors]
Now, if for any value of x g(x)=0
Then, f(x) will also be equal to zero.
Thus we get a theorem to solve your question.
If you know the Factor Theorem, The Real Answer Starts HERE:
Now that you know the key theorem this question becomes pretty easy.
Here, f(x)= x^3 + ax^2- 2x + a + 4
And, g(x)= x + a
We have, x + a = 0
=> x = -a
Now, put the value thus found into f(x) function
f(-a) = (-a)^3 + a(-a)^2 - 2(-a) + a + 4 =0
=> -a^3 + a^3 + 2a + a = -4
=> 3a = -4
=> a = -4/3