Remainder/factor theorem question

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In summary, the question is asking for a method to solve for a and b when given f(x) and the divisor, x2 - 5x +6. One possible solution is to use the long division method or inspection to determine the values of a, b, c, and d.
  • #1
HerroFish
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The question is
Find "a" and "b":

f(x) is
ax^3 - bx^2 + 2x - 12

Divisor is
x^2 - 5x +6

Remainder is
2x - 3So what I tried to do was divide f(x) with the divisor and take the the remainder that I got and let it = 2x - 3 but I always get stuck in the end. My teacher suggested using the division statement but I'm not quite sure how that will help.

Thanks for helping!
 
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  • #2
One method to solve this is use polynomial long division starting at the end, where you know the remainder, the divisor, and the last two terms of the dividend, which allows you to determine the last term of the quotient, then work backwards until you solve for b then a.
 
  • #3
ouu okay I see!
But is there a shorter method to this?

Thanks in advance
 
  • #4
What you have is

ax^3 - bx^2 + 2x - 12 = (cx + d) (x^2 - 5x + 6) + (2x - 3)

where a, b, c and d are unknown. You could subract (2x - 3) from both sides first, then use the long division method or inspection to determine the values, but subtracting (2x - 3) from both sides is the same as what you're doing in the first step of working the long division backwards. This backwards long division method is used in general for "division problems" when there are unknowns in the dividend, divisor, and/or remainder.
 
Last edited:
  • #5
HerroFish said:
The question is
Find "a" and "b":

f(x) is
ax^3 - bx^2 + 2x - 12

Divisor is
x^2 - 5x +6

Remainder is
2x - 3

So what I tried to do was divide f(x) with the divisor and take the the remainder that I got and let it = 2x - 3 but I always get stuck in the end. My teacher suggested using the division statement but I'm not quite sure how that will help.

Thanks for helping!
So, If f(x) divided by g(x) yields a quotient q(x) and remainder r(x), then

[itex]f(x)=g(x)\cdot q(x)+r(x)\ .[/itex]

Therefore, [itex]f(x)-r(x)=g(x)\cdot q(x)\ .[/itex] So that g(x) divides (f(x)-r(x)) with a remainder of zero.

Notice that in your problem the divisor, x2 - 5x +6, is factorable. Apply the factor theorem to f(x)-r(x), with each of those factors.
 
  • #6
:biggrin:
SammyS said:
So, If f(x) divided by g(x) yields a quotient q(x) and remainder r(x), then

[itex]f(x)=g(x)\cdot q(x)+r(x)\ .[/itex]

Therefore, [itex]f(x)-r(x)=g(x)\cdot q(x)\ .[/itex] So that g(x) divides (f(x)-r(x)) with a remainder of zero.

Notice that in your problem the divisor, x2 - 5x +6, is factorable. Apply the factor theorem to f(x)-r(x), with each of those factors.

Okay I get it now!
Thanks everybody for all your help! :smile: :!)
 

1. What is the Remainder Theorem?

The Remainder Theorem is a mathematical theorem that states that if a polynomial function f(x) is divided by (x-a), then the remainder is equal to f(a). In simpler terms, it tells us that when we divide a polynomial by (x-a), the remainder will be the value of the polynomial when x is equal to a.

2. How is the Remainder Theorem related to the Factor Theorem?

The Factor Theorem is closely related to the Remainder Theorem. It states that if a polynomial function f(x) is divided by (x-a) and the remainder is equal to 0, then (x-a) is a factor of f(x). This means that the Factor Theorem helps us determine if a given value is a root (zero) of a polynomial function, and the Remainder Theorem tells us what the remainder will be when dividing by that root.

3. How do you use the Remainder Theorem to find the remainder of a polynomial?

To use the Remainder Theorem, you first need to divide the polynomial function by (x-a) using long or synthetic division. The last term in the resulting quotient will be the remainder, which can also be written as f(a) where f(x) is the original polynomial function. Alternatively, you can use the remainder formula, which states that the remainder is equal to f(a)/g(a), where g(a) is the coefficient of x-a in the quotient.

4. Can the Remainder Theorem be used for polynomials with multiple variables?

No, the Remainder Theorem can only be used for polynomials with one variable. This is because the theorem relies on plugging in a specific value for x, which is not possible when there are multiple variables. However, there are similar theorems for polynomials with multiple variables, such as the Multivariate Remainder Theorem.

5. How is the Factor Theorem used to find the roots of a polynomial?

The Factor Theorem is used to find the roots (zeros) of a polynomial by first determining if a given value is a root. If (x-a) is a factor of the polynomial, then a is a root. This can be done by using long or synthetic division, or by plugging in the value of a into the polynomial function and checking if the result is equal to 0. Once a root is found, it can be factored out, and the process can be repeated until all roots are found.

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