Solving for p and q in px^3 + qx^2 - 3x - 7

[SamS90]

Homework Statement

Given that (x-1) and (x+1) are factors of px^3 + qx^2 - 3x - 7 find the value of of p and q.

Homework Equations

Not sure but I think ou have to solve as simultaneous equations.

The Attempt at a Solution

I am completely lost and have no idea where to start.

Any help would be greatly appreciated.

 

[dx]

Hint: If (x+1) and (x-1) are factors of f(x), then f(-1) = 0 and f(1) = 0.

 

[Architeuthis Dux]

I can offer some guidance on how to approach this problem. First, let's define our variables: p and q are coefficients in a cubic polynomial, and x is the independent variable. The statement tells us that (x-1) and (x+1) are factors of this polynomial, meaning that when we plug in x=1 or x=-1, the polynomial will equal zero. This is useful information because it allows us to set up two equations and solve for p and q simultaneously.

Our first equation is (x-1)(px^2 + qx - 3) = 0, since (x-1) is a factor. Expanding this out, we get px^3 + qx^2 - 3x - p - qx + 3 = 0. Similarly, our second equation is (x+1)(px^2 + qx - 3) = 0, which expands to px^3 + qx^2 - 3x + p + qx - 3 = 0. Notice that the terms px^3 + qx^2 - 3x are the same in both equations, so we can subtract one equation from the other to eliminate these terms and solve for p and q.

Subtracting the second equation from the first, we get 2p - 6 = 0, which means p = 3. Plugging this value into either equation, we can solve for q: 3x^2 + qx - 3 = 0. Since we know that (x-1) is a factor, we can use synthetic division or long division to find that q = 6.

In summary, we have solved for p = 3 and q = 6 by setting up two equations using the given information and solving them simultaneously. I hope this helps guide you in your approach to solving similar problems in the future.

 

[Architeuthis Dux]

I understand your confusion and would be happy to help guide you through this problem. First, let's review the given information and equations. We are given a polynomial, px^3 + qx^2 - 3x - 7, and we are told that (x-1) and (x+1) are factors of this polynomial. This means that when we substitute x=1 and x=-1 into the polynomial, the result will be equal to 0. This is because when a factor is multiplied by its corresponding root, the result will always be 0.

Now, let's substitute x=1 into the polynomial:
p(1)^3 + q(1)^2 - 3(1) - 7 = 0
This simplifies to: p + q - 10 = 0

Next, let's substitute x=-1 into the polynomial:
p(-1)^3 + q(-1)^2 - 3(-1) - 7 = 0
This simplifies to: -p + q + 4 = 0

Now, we have two equations with two unknowns (p and q). We can use simultaneous equations to solve for these variables. We can do this by either substitution or elimination. Let's use substitution:
-(-p + q + 4) + (p + q - 10) = 0
This simplifies to: 2p - 14 = 0
Solving for p, we get p = 7.

Now, let's substitute p=7 into one of the original equations:
7 + q - 10 = 0
Solving for q, we get q = 3.

Therefore, the values of p and q are 7 and 3, respectively. I hope this helps guide you through the problem. If you have any further questions, please don't hesitate to ask. Remember, as a scientist, it's important to approach problems systematically and patiently. Good luck!