Finding the value of unknown coefficients using a known factor

Nov 2012
17
0
Ireland
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)
 

topsquark

Forum Staff
Jan 2006
11,602
3,457
Wellsville, NY
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 2012
17
0
Ireland
Cheers Dan. Can you explain what you mean in a bit more detail? Dividing the two what?

Thanks.
 

topsquark

Forum Staff
Jan 2006
11,602
3,457
Wellsville, NY
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
 
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