Finding the value of unknown coefficients using a known factor

• Nov 4th 2012, 04:52 PM
indymogul
Finding the value of unknown coefficients using a known factor
This one is really wrecking my head. I've been working on this for hours and I just can't seem to get the correct answer. I'm starting to think there is an error in the textbook, but I've been studying for hours and I think I may just be losing my focus.

The question is:

Find the value of p and the value of q if px^3 + qx^2 - 58x -15 is divisible by x^2 + 2x -15.

So, I've broken the factor down into roots, namely (x + 5) and (x - 3). Using the factor theorem, if f(x) = px^3 + qx^2 - 58x -15, then f(-5) and f(3) should both be equal to 0, right?

However I've tried and tried and I just can't get the correct values for p an q, which are given as 4 and 9, respectively.

p(-5)^3 + q(-5)^2 - 58(-5) -15 = 0

p(-125) + q(25) + 290 -15 = 0

-125p + 25q + 275 = 0

25q + 275 = 125p

q + 11 = 5p

q = 5p - 11

************************
p(3)^3 + q(3)^2 - 58(3) -15 = 0

27p + 9q - 174 - 15 = 0

27p + 9q - 189 = 0

27p = 189 - 9q

3p = 21 - q

p = 7 - q/3

now, express q in terms of p

p = 7 - (5p - 11)/3

3p = 21 - 5p -11

8p = 10 or p = 10/8 or 1.25. But p should equal 4???

I have done this over and over and over and over and I just can't get the correct values for p and q. Can someone please tell me where I'm going wrong?

Thank you. You will be saving me from total brain meltdown. (Headbang)
• Nov 4th 2012, 05:24 PM
topsquark
Re: Finding the value of unknown coefficients using a known factor
Quote:

Originally Posted by indymogul
This one is really wrecking my head. I've been working on this for hours and I just can't seem to get the correct answer. I'm starting to think there is an error in the textbook, but I've been studying for hours and I think I may just be losing my focus.

The question is:

Find the value of p and the value of q if px^3 + qx^2 - 58x -15 is divisible by x^2 + 2x -15.

So, I've broken the factor down into roots, namely (x + 5) and (x - 3). Using the factor theorem, if f(x) = px^3 + qx^2 - 58x -15, then f(-5) and f(3) should both be equal to 0, right?

However I've tried and tried and I just can't get the correct values for p an q, which are given as 4 and 9, respectively.

p(-5)^3 + q(-5)^2 - 58(-5) -15 = 0

p(-125) + q(25) + 290 -15 = 0

-125p + 25q + 275 = 0

25q + 275 = 125p

q + 11 = 5p

q = 5p - 11

************************
p(3)^3 + q(3)^2 - 58(3) -15 = 0

27p + 9q - 174 - 15 = 0

27p + 9q - 189 = 0

27p = 189 - 9q

3p = 21 - q

p = 7 - q/3

now, express q in terms of p

p = 7 - (5p - 11)/3

3p = 21 - 5p -11

8p = 10 or p = 10/8 or 1.25. But p should equal 4???

I have done this over and over and over and over and I just can't get the correct values for p and q. Can someone please tell me where I'm going wrong?

Thank you. You will be saving me from total brain meltdown. (Headbang)

I can't immediately see the problem with your work, but have you considered simply dividing the two? It's a bit messy but I was able to get the solution with little trouble. (p = 4 and q = 9).

-Dan
• Nov 4th 2012, 05:32 PM
indymogul
Re: Finding the value of unknown coefficients using a known factor
Cheers Dan. Can you explain what you mean in a bit more detail? Dividing the two what?

Thanks.
• Nov 4th 2012, 06:32 PM
topsquark
Re: Finding the value of unknown coefficients using a known factor
Quote:

Originally Posted by indymogul
Cheers Dan. Can you explain what you mean in a bit more detail? Dividing the two what?

Thanks.

Good old fashioned long division. (px^3 + qx^2 - 58x - 15)/(x^2 + 2x - 15). You'll have to keep track of the "p's and q's" in order to work it down. Your remainder line (your last line) will look like [-2(q - 2p) + 15p- 58]x + 15(q - 2p - 1). In order to make the two polynomials divide with no remainder you set both coefficients to zero. Two equations, two unknowns.

-Dan