Find the values of a and b in the given polynomial

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This is getting a bit silly, but anyway. The statement$$3n^2-3n+1=3k^2-3n^2$$is not true. So you cannot infer from n^2 = k^2 that 3n^2 = 3k^2. You can infer from n = k that 3n = 3k. Also, you don't have to make k = n - 1. You can choose any value for k you like, as long as it is a constant value. If you choose k = 2n then 3k = 6n and if you choose k = n/2 then 3
  • #1
chwala
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Homework Statement
Find the value of ##a## and ##b## in terms of ##n## given;
##(x-n+1)^3-(x-n)^3=3x^2+ax+b##
Relevant Equations
Binomial theorem
My approach;

Let ##(n+1)=k##
##(x-n+1)^3-(x-n)^3=3x^2+ax+b##
##(x-k)^3-(x-n)^3=3x^2+ax+b##
##(x^3-3x^2k+3xk^2-k^3)-(x^3-3x^2n+3xn^2-n^3)##
##=-3x^2k+3x^2n+3xk^2-3xn^2-k^3+n^3≡3x^2+ax+b##

##⇒1=n-k##
##⇒3(k^2-n^2)=a##
##⇒n^3-k^3=b##

##3(k+n)(k-n)=a##
##-3(k+n)=a##
##a=-3(n-1+n)##
##a=-3(2n-1)##
##a=3-6n##

##b=n^3-k^3=n^3-(n-1)^3=n^3-n^3+3n^2-3n+1=3n^2-3n+1##
##b=3n^2-3n+1##

which agrees with textbook solution...i would appreciate alternative approach.
 
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  • #2
Let x=0 on the both sides
[tex]b=n^3-(n-1)^3=n^2+n(n-1)+(n-1)^2=3n^2-3n+1[/tex]

Then let x=n on the both sides
[tex]0=6n^2+(a-3)n[/tex]
[tex]a=-6n+3[/tex]
 
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  • #3
chwala said:
Homework Statement:: Find the value of ##a## and ##b## in terms of ##n## given;
##(x-n+1)^3-(x-n)^3=3x^2+ax+b##
Relevant Equations:: Binomial theorem

My approach;

Let ##(n+1)=k##
##(x-n+1)^3-(x-n)^3=3x^2+ax+b##
##(x-k)^3-(x-n)^3=3x^2+ax+b##
You have a mistake almost at the very beginning. x - n + 1 = x - (n - 1), so a more reasonable substitution would be to let k = n - 1.
In your work, with your substitution, ##(x - k)^3 = (x - (n + 1))^3 = (x - n - 1)^3##. This isn't what you started with.
chwala said:
##(x^3-3x^2k+3xk^2-k^3)-(x^3-3x^2n+3xn^2-n^3)##
##=-3x^2k+3x^2n+3xk^2-3xn^2-k^3+n^3≡3x^2+ax+b##

##⇒1=n-k##
##⇒3(k^2-n^2)=a##
##⇒n^3-k^3=b##

##3(k+n)(k-n)=a##
##-3(k+n)=a##
##a=-3(n-1+n)##
##a=-3(2n-1)##
##a=3-6n##

##b=n^3-k^3=n^3-(n-1)^3=n^3-n^3+3n^2-3n+1=3n^2-3n+1##
##b=3n^2-3n+1##

which agrees with textbook solution...i would appreciate alternative approach.
 
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  • #4
Mark44 said:
You have a mistake almost at the very beginning. x - n + 1 = x - (n - 1), so a more reasonable substitution would be to let k = n - 1.
In your work, with your substitution, ##(x - k)^3 = (x - (n + 1))^3 = (x - n - 1)^3##. This isn't what you started with.
I see your point, what about letting ##k=n+1?## and not ##k=(n+1).## I can see that it's the bracket that causes all the confusion.
 
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  • #5
chwala said:
which agrees with textbook solution...i would appreciate alternative approach.
I don't see how you arrived at the same solution as the textbook, given the mistake I pointed out. I didn't go through your work beyond what I pointed out, but I suspect that you have made another mistake somewhere.
 
  • #6
He says at first line n+1=k but all the rest lines are working like n-1=k, that's why he arrived at the right solution. So practically only his first line is wrong.
 
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  • #7
anuttarasammyak said:
Let x=0 on the both sides
...
f(x) is a polynomial. Its constant term is f(0).

anuttarasammyak said:
Then let x=n on the both sides
...
Alternatively, the coefficient of x is f'(0). That's not pre-calc, but he did ask for an alternative solution.
 
  • #8
Prof B said:
f(x) is a polynomial. Its constant term is f(0).Alternatively, the coefficient of x is f'(0). That's not pre-calc, but he did ask for an alternative solution.
@Prof B I do not think you have indicated any improvement on the problem nor alternative solution. Can you show your working by using ##f'(0)## as you have pointed out?
 
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  • #9
Try expanding the cubic term with binomial theorem like this
$$
((x-n)+1)^3=(x-n)^3 + 3(x-n)^2 + 3(x-n) +1
$$
and observe that the ##(x-n)^3## gets subtracted off on the r.h.s. In other words; think ahead. I think that was the point of this exercise.
 
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  • #10
chwala said:
@Prof B I do not think you have indicated any improvement on the problem nor alternative solution. Can you show your working by using ##f'(0)## as you have pointed out?
f'(0) is the derivative of f(x) evaluated at 0. It will make sense when you take calculus.
 
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  • #11
Prof B said:
f'(0) is the derivative of f(x) evaluated at 0. It will make sense when you take calculus.
He knows calculus, he is 43 years old :wink:. You can check additional info for each member, either by clicking at the avatar or hovering the mouse over it. Just saying cause you are new at these forums.
And welcome to PF from me, first indication is that you are quite knowledgeable about math!
 
  • #12
I'm not sure how I was supposed to infer that. Pythagoras was a great mathematician. He lived to the age of 75 and did not know calculus.
 
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  • #13
Prof B said:
I'm not sure how I was supposed to infer that. Pythagoras was a great mathematician. He lived to the age of 75 and did not know calculus.
Yes ok I understand that. You don't know @chwala, he is quite sometime in these forums and has post many problems over a wide range of math topics, including calculus.
 
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  • #14
chwala said:
Homework Statement:: Find the value of ##a## and ##b## in terms of ##n## given;
##(x-n+1)^3-(x-n)^3=3x^2+ax+b##
Relevant Equations:: Binomial theorem

My approach;

Let ##(n+1)=k##
The equation above is wrong. You want to replace x - n + 1 by x - k, so k = n - 1.
chwala said:
##(x-n+1)^3-(x-n)^3=3x^2+ax+b##
##(x-k)^3-(x-n)^3=3x^2+ax+b##
##(x^3-3x^2k+3xk^2-k^3)-(x^3-3x^2n+3xn^2-n^3)##
##=-3x^2k+3x^2n+3xk^2-3xn^2-k^3+n^3≡3x^2+ax+b##

##⇒1=n-k##
This isn't equivalent to n + 1 = k, which was the substitution you made several lines up.
chwala said:
##⇒3(k^2-n^2)=a##
##⇒n^3-k^3=b##
In addition to the comments above, the work above seems incomplete. The 2nd and 3rd equations above come from equating the x terms and constant terms on both sides of the equation above the one that says 1 = n - k. The equation that seems to be missing is the one that equates the ##x^2## terms on both sides. Once I had these three equations, then I would solve for a and b.

chwala said:
##3(k+n)(k-n)=a##
##-3(k+n)=a##
Here's another mistake that, fortunately for you, cancels out your earlier mistake. According to your substitution (which is incorrect), k = n + 1. This means, with that substitution, that the 2nd equation above should be ##3(k + n)(n + 1 - n) = a##, or ##3(k + n)(1)##.

chwala said:
##a=-3(n-1+n)##
##a=-3(2n-1)##
##a=3-6n##

##b=n^3-k^3=n^3-(n-1)^3=n^3-n^3+3n^2-3n+1=3n^2-3n+1##
##b=3n^2-3n+1##

which agrees with textbook solution...i would appreciate alternative approach.
 
  • #15
@Mark44 hi, let me look at it again...Will get back on this.
 
  • #16
Ok here we have it;...pretty easy

##x-n+1)^3-(x-n)^3=3x^2+ax+b##

Re-arranging;

##(x+1-n)^3-(x-n)^3=3x^2+ax+b##

Using Combinations-

##[^3C_0x^3+3C_1x^2 (1-n)+3C_2x (1-n)^2 +3C_3x^0 (1-n)^3]-[3C_0x^3+3C_1x^2(-n)+3C_2x(-n)^2+3C_3x^0 (-n)^3]##

##=[x^3+3x^2(1-n)+3x(1-n)^2+(1-n)^3]-[x^3-3x^2n+3xn^2-n^3]##

##=[x^3+3x^2-3x^2n+3x-6xn+3xn^2-n^3+3n^2-3n+1]-[x^3-3x^2n+3xn^2-n^3]##

##=3x^2+3x-6xn+3n^2-3n+1##

##=3x^2+(3-6n)x+3n^2-3n+1##

therefore,

##a=3-6n##

##b=3n^2-3n+1##
 

FAQ: Find the values of a and b in the given polynomial

What is a polynomial?

A polynomial is a mathematical expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication. It can have one or more terms, and the degree of a polynomial is determined by the highest exponent of the variable.

How do I find the values of a and b in a polynomial?

To find the values of a and b in a polynomial, you can use a system of equations. Set the polynomial equal to a given value and solve for the variables a and b. You can also use substitution or elimination methods to solve for the variables.

Can I use the quadratic formula to find the values of a and b in a polynomial?

Yes, you can use the quadratic formula to find the values of a and b in a polynomial. The quadratic formula is used to solve quadratic equations, which are a type of polynomial with a degree of 2. By setting the polynomial equal to 0 and using the quadratic formula, you can solve for the values of a and b.

What is the difference between a and b in a polynomial?

In a polynomial, a and b are variables that represent coefficients. The variable a typically represents the coefficient of the term with the highest degree, while b represents the coefficient of the term with the next highest degree. These variables are used to find the values of the coefficients in a polynomial expression.

Are there any other methods for finding the values of a and b in a polynomial?

Yes, there are other methods for finding the values of a and b in a polynomial. These include factoring, graphing, and using the rational root theorem. Depending on the specific polynomial, one method may be more efficient or accurate than others.

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