Westonci.ca is your trusted source for accurate answers to all your questions. Join our community and start learning today! Join our platform to connect with experts ready to provide detailed answers to your questions in various areas. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
To find the values of [tex]\( m \)[/tex] and [tex]\( n \)[/tex] such that the polynomial [tex]\( 2x^3 + mx^2 + nx - 14 \)[/tex] is exactly divisible by [tex]\( x^2 + x - 2 \)[/tex], we need to ensure that the polynomial [tex]\( 2x^3 + mx^2 + nx - 14 \)[/tex] has [tex]\( x^2 + x - 2 \)[/tex] as a factor. Here are the steps:
1. Factor the divisor:
[tex]\[ x^2 + x - 2 = (x - 1)(x + 2) \][/tex]
This means that the roots of [tex]\( x^2 + x - 2 \)[/tex] are [tex]\( x = 1 \)[/tex] and [tex]\( x = -2 \)[/tex].
2. Setting up conditions for the polynomial:
For [tex]\( 2x^3 + mx^2 + nx - 14 \)[/tex] to be divisible by [tex]\( x^2 + x - 2 \)[/tex], it must be zero when [tex]\( x = 1 \)[/tex] and [tex]\( x = -2 \)[/tex].
3. Substitute [tex]\( x = 1 \)[/tex] into the polynomial:
[tex]\[ P(1) = 2(1)^3 + m(1)^2 + n(1) - 14 = 2 + m + n - 14 \][/tex]
Simplify to:
[tex]\[ 2 + m + n - 14 = 0 \quad \Rightarrow \quad m + n - 12 = 0 \][/tex]
So the first equation is:
[tex]\[ m + n = 12 \][/tex]
4. Substitute [tex]\( x = -2 \)[/tex] into the polynomial:
[tex]\[ P(-2) = 2(-2)^3 + m(-2)^2 + n(-2) - 14 = 2(-8) + m(4) + n(-2) - 14 \][/tex]
Simplify to:
[tex]\[ -16 + 4m - 2n - 14 = 0 \quad \Rightarrow \quad 4m - 2n - 30 = 0 \][/tex]
So the second equation is:
[tex]\[ 4m - 2n = 30 \][/tex]
5. Solve the system of equations:
[tex]\[ \begin{cases} m + n = 12 \\ 4m - 2n = 30 \end{cases} \][/tex]
First, solve the second equation for [tex]\( n \)[/tex]:
[tex]\[ 4m - 2n = 30 \quad \Rightarrow \quad 2m - n = 15 \quad \Rightarrow \quad n = 2m - 15 \][/tex]
Substitute [tex]\( n = 2m - 15 \)[/tex] into the first equation:
[tex]\[ m + (2m - 15) = 12 \][/tex]
Simplify:
[tex]\[ 3m - 15 = 12 \quad \Rightarrow \quad 3m = 27 \quad \Rightarrow \quad m = 9 \][/tex]
Substitute [tex]\( m = 9 \)[/tex] back into [tex]\( n = 2m - 15 \)[/tex]:
[tex]\[ n = 2(9) - 15 = 18 - 15 = 3 \][/tex]
6. Conclude the result:
The values are [tex]\( m = 9 \)[/tex] and [tex]\( n = 3 \)[/tex]. Therefore, [tex]\( m + n = 9 + 3 = 12 \)[/tex].
Thus, the value of [tex]\( m + n \)[/tex] is [tex]\(\boxed{12}\)[/tex].
1. Factor the divisor:
[tex]\[ x^2 + x - 2 = (x - 1)(x + 2) \][/tex]
This means that the roots of [tex]\( x^2 + x - 2 \)[/tex] are [tex]\( x = 1 \)[/tex] and [tex]\( x = -2 \)[/tex].
2. Setting up conditions for the polynomial:
For [tex]\( 2x^3 + mx^2 + nx - 14 \)[/tex] to be divisible by [tex]\( x^2 + x - 2 \)[/tex], it must be zero when [tex]\( x = 1 \)[/tex] and [tex]\( x = -2 \)[/tex].
3. Substitute [tex]\( x = 1 \)[/tex] into the polynomial:
[tex]\[ P(1) = 2(1)^3 + m(1)^2 + n(1) - 14 = 2 + m + n - 14 \][/tex]
Simplify to:
[tex]\[ 2 + m + n - 14 = 0 \quad \Rightarrow \quad m + n - 12 = 0 \][/tex]
So the first equation is:
[tex]\[ m + n = 12 \][/tex]
4. Substitute [tex]\( x = -2 \)[/tex] into the polynomial:
[tex]\[ P(-2) = 2(-2)^3 + m(-2)^2 + n(-2) - 14 = 2(-8) + m(4) + n(-2) - 14 \][/tex]
Simplify to:
[tex]\[ -16 + 4m - 2n - 14 = 0 \quad \Rightarrow \quad 4m - 2n - 30 = 0 \][/tex]
So the second equation is:
[tex]\[ 4m - 2n = 30 \][/tex]
5. Solve the system of equations:
[tex]\[ \begin{cases} m + n = 12 \\ 4m - 2n = 30 \end{cases} \][/tex]
First, solve the second equation for [tex]\( n \)[/tex]:
[tex]\[ 4m - 2n = 30 \quad \Rightarrow \quad 2m - n = 15 \quad \Rightarrow \quad n = 2m - 15 \][/tex]
Substitute [tex]\( n = 2m - 15 \)[/tex] into the first equation:
[tex]\[ m + (2m - 15) = 12 \][/tex]
Simplify:
[tex]\[ 3m - 15 = 12 \quad \Rightarrow \quad 3m = 27 \quad \Rightarrow \quad m = 9 \][/tex]
Substitute [tex]\( m = 9 \)[/tex] back into [tex]\( n = 2m - 15 \)[/tex]:
[tex]\[ n = 2(9) - 15 = 18 - 15 = 3 \][/tex]
6. Conclude the result:
The values are [tex]\( m = 9 \)[/tex] and [tex]\( n = 3 \)[/tex]. Therefore, [tex]\( m + n = 9 + 3 = 12 \)[/tex].
Thus, the value of [tex]\( m + n \)[/tex] is [tex]\(\boxed{12}\)[/tex].
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.