Polynomials Problem:
1. List two factors of h(x) of the form x−r: x, x + 2 (right answer)
2. A function of the form ax^n that approximates h(x) for large values of |x| is f(x)= 2x^4. (right answer)
3. This tells us that the degree of h is 4 and the coefficient of the highest degree term of h(x)is 2.(4&2 right answer)
4. If (x2+1) is also a factor of h(x), then h(x)= _______________ ?
Could someone please help me get the 4th question? I have tried many times but can't get it :(
It would depend on what h(x) is. You did not list it.
The question didn't list h(x) that's what we are trying to figure out using the all the information above (1-3).
If (x^2+1) is also a factor of h(x), then h(x) = ____________ ?
If that's all we're given, I do not understand how you got the answer for question a, namely the factors of h(x) correctly as x and (x+2).
If x, x+2 and (x²+1) are factors of h(x), then
h(x)=2x(x+2)(x²+1)
since the highest term is 2x².
Thank You so much MathMate, the answer you gave is right :)
how can i get the 2x^4 for largest value of |x| i don't understand
To find the expression for h(x) if (x^2+1) is a factor, we can use polynomial division. Here's how:
Step 1: Set up the polynomial division:
Divide (x^2+1) into h(x), which gives us:
_______________________
(x^2+1) | h(x)
Step 2: Divide the highest degree terms:
Divide x^4 (from h(x)) by x^2, which gives us x^2.
x^2
_______________________
(x^2+1) | h(x)
Step 3: Multiply the divisor (x^2+1) by the quotient (x^2):
Multiply x^2 by (x^2+1), which gives us x^4+x^2.
x^2
_______________________
(x^2+1) | h(x)
- (x^4 + x^2)
Step 4: Subtract the result of the multiplication from h(x):
Subtract x^4+x^2 from h(x), which gives us the remainder.
x^2
_______________________
(x^2+1) | h(x)
- (x^4 + x^2)
________________
...
Step 5: Repeat Steps 2-4 until you can no longer divide:
Continue dividing until you reach the remainder with a degree less than the divisor (x^2+1). This is when you can no longer divide evenly.
x^2 + ?
_______________________
(x^2+1) | h(x)
- (x^4 + x^2)
________________
...
Step 6: Write the expression for h(x):
The remainder obtained in Step 5 will be the expression for h(x) when (x^2+1) is a factor.
So, the expression for h(x) is the remainder obtained when dividing h(x) by (x^2+1).