If p(x) is a polynomial and is divided by (x-k) and a remainder is obtained, then that remainder is p(k). If the quadratic p(x)=x^2-3x+5 gives the same remainder when divided by x+k as it does when divided by x-3k find the value of k, k is not equal to 0.

This is the answer which was in my homework book, though I don't quite get it:
p(-k)= k^2+3k+5
p(3k)= 9k^2-9k+5
Equating k^2+3k+5=9k^2-9k+5;
8k^2-12k=0
4k(2k-3)=0
Ignoring k=0, so k=1.5

Can someone please provide a clearer explanation to this problem?

what don't you get? Divide by x+k is the same as divide by x-(-k), so the remainder is p(-k). Same for p(3k).

They said the remainders are equal. So, equate the two expressions and solve for k.

To check, just evaluate p(x).

p(-3/2) = p(9/2) = 47/4

Sure, let me try to explain it to you in a clearer way.

We know that when we divide a polynomial p(x) by (x - k), we get a remainder that is equal to p(k). So, when we divide the quadratic p(x) = x^2 - 3x + 5 by (x + k), the remainder we obtain will be equal to p(-k).

So, when we divide p(x) by (x - 3k), the remainder we obtain will be equal to p(3k). We can set up the equations as follows:

p(-k) = k^2 + 3k + 5 (Using the remainder theorem)
p(3k) = (3k)^2 - 3(3k) + 5 (Using the remainder theorem)

Now we equate these two values to find the value of k.

k^2 + 3k + 5 = 9k^2 - 9k + 5

Simplifying this equation, we get:

8k^2 - 12k = 0

Factoring out 4k, we have:

4k(2k - 3) = 0

Ignoring the solution k = 0 (as it was stated in the problem), we are left with:

2k - 3 = 0

Solving this equation, we find:

2k = 3
k = 3/2
k = 1.5

So the value of k, which is not equal to 0, is 1.5.

I hope this explanation helps you better understand the problem! Let me know if you have any further questions.

To understand this problem, let's break it down step-by-step:

1. Start with the given quadratic polynomial, p(x) = x^2 - 3x + 5.

2. Assume that when p(x) is divided by (x + k), it gives the same remainder as when divided by (x - 3k).

3. When p(x) is divided by (x + k), the remainder can be represented as p(-k). This is because the remainder is obtained by substituting x = -k into p(x).

4. Similarly, when p(x) is divided by (x - 3k), the remainder can be represented as p(3k). This is because the remainder is obtained by substituting x = 3k into p(x).

5. Now, equate the remainders obtained from both divisions: p(-k) = p(3k).

6. Substitute the polynomial p(x) = x^2 - 3x + 5 into the equation from step 5:

p(-k) = (-k)^2 - 3(-k) + 5
= k^2 + 3k + 5

p(3k) = (3k)^2 - 3(3k) + 5
= 9k^2 - 9k + 5

7. We have the equation: k^2 + 3k + 5 = 9k^2 - 9k + 5.

8. Simplify the equation: 8k^2 - 12k = 0.

9. Factor out k: 4k(2k - 3) = 0.

10. Solve for k by setting each factor equal to zero:

a) k = 0 (ignore this solution, as it is stated that k is not equal to 0)
b) 2k - 3 = 0
2k = 3
k = 3/2 = 1.5

11. Therefore, the value of k that satisfies the given conditions is k = 1.5.

Sure! Let's break down the problem step by step to understand it better.

First, let's assume that the quadratic polynomial p(x) = x^2 - 3x + 5 gives the same remainder when divided by x + k as it does when divided by x - 3k.

When we divide a polynomial by (x - k), the remainder is given by substituting k into the polynomial. In other words, if we divide p(x) by (x - k) and get a remainder R(x), then R(k) would be equal to the remainder.

So, let's calculate the remainder when p(x) = x^2 - 3x + 5 is divided by (x + k). To do this, we substitute -k into the polynomial:

p(-k) = (-k)^2 - 3(-k) + 5
= k^2 + 3k + 5 (Equation 1)

Similarly, let's calculate the remainder when p(x) is divided by (x - 3k). We substitute 3k into the polynomial:

p(3k) = (3k)^2 - 3(3k) + 5
= 9k^2 - 9k + 5 (Equation 2)

Now, according to the problem, the remainder obtained from both divisions is the same. So we equate Equation 1 and Equation 2:

k^2 + 3k + 5 = 9k^2 - 9k + 5

Simplifying this equation, we get:

8k^2 - 12k = 0

Factoring out 4k, we have:

4k(2k - 3) = 0

Now, we have two possible solutions: 4k = 0 or 2k - 3 = 0.

Since the problem states that k is not equal to 0, we ignore the first solution. We solve the second equation:

2k - 3 = 0
2k = 3
k = 3/2 = 1.5

So, the value of k that satisfies the condition is k = 1.5.

I hope this explanation clarifies the problem for you! Let me know if you have any further questions.