Get detailed and accurate answers to your questions on IDNLearn.com. Our platform offers reliable and detailed answers, ensuring you have the information you need.

If the polynomial [tex]\( p(x) = x^3 - (m-2)x^2 - mx + 28 \)[/tex] is divided by [tex]\((x+3)\)[/tex] and the remainder is 10, find the value of [tex]\( m \)[/tex].

Sagot :

Certainly! Let's solve this step-by-step.

Given the polynomial:
[tex]\[ p(x) = x^3 - (m-2)x^2 - mx + 28 \][/tex]

We know that when this polynomial is divided by [tex]\( (x+3) \)[/tex], the remainder is 10. By the Remainder Theorem, the remainder when [tex]\( p(x) \)[/tex] is divided by [tex]\( (x+3) \)[/tex] is [tex]\( p(-3) \)[/tex].

So, let's plug [tex]\( x = -3 \)[/tex] into the polynomial and set it equal to 10:

[tex]\[ p(-3) = (-3)^3 - (m-2)(-3)^2 - m(-3) + 28 = 10 \][/tex]

Now, let's evaluate each term separately:

1. [tex]\( (-3)^3 = -27 \)[/tex]
2. [tex]\( (-3)^2 = 9 \)[/tex]
3. So, [tex]\( -(m-2)(-3)^2 = -(m-2) \cdot 9 = -9(m-2) \)[/tex]
4. And, [tex]\( -m(-3) = 3m \)[/tex]

Substituting these into the polynomial:

[tex]\[ -27 - 9(m-2) + 3m + 28 = 10 \][/tex]

Now, let's simplify the equation:

[tex]\[ -27 - 9m + 18 + 3m + 28 = 10 \][/tex]

Combine the like terms:

[tex]\[ -27 + 18 + 28 - 9m + 3m = 10 \][/tex]
[tex]\[ 19 - 6m = 10 \][/tex]

To isolate [tex]\( m \)[/tex], we solve the equation:

[tex]\[ 19 - 6m = 10 \][/tex]
[tex]\[ -6m = 10 - 19 \][/tex]
[tex]\[ -6m = -9 \][/tex]

Now, divide both sides by -6:

[tex]\[ m = \frac{-9}{-6} \][/tex]
[tex]\[ m = \frac{3}{2} \][/tex]

Therefore, the value of [tex]\( m \)[/tex] is:

[tex]\[ m = 1.5 \][/tex]
Thank you for using this platform to share and learn. Keep asking and answering. We appreciate every contribution you make. For trustworthy answers, rely on IDNLearn.com. Thanks for visiting, and we look forward to assisting you again.