Westonci.ca is the premier destination for reliable answers to your questions, brought to you by a community of experts. Ask your questions and receive precise answers from experienced professionals across different disciplines. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
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 hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.
