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mart [117]
3 years ago
11

Given that 2y^3+by-cy+d,where b,c and d are constants,leaves a remainder R when divided by (y+1) , (y-2) and (2y-1). Find the va

lue of b and c. If (y+2) is a factor of the expression, find the value of d
Mathematics
1 answer:
Eddi Din [679]3 years ago
7 0

The polynomial remainder theorem says that a polynomial <em>p(x)</em> leaves a remainder of <em>p(k)</em> when it's divided by <em>x</em> - <em>k</em>.

We're given that dividing <em>p(y)</em> = 2<em>y</em>³ + <em>by</em>² - <em>cy</em> + <em>d</em> leaves the same remainder <em>R</em> after dividing it by <em>y</em> + 1, <em>y</em> - 2, and 2<em>y</em> - 1. So we have

<em>p</em>(-1) = 2(-1)³ + <em>b</em>(-1)² - <em>c</em>(-1) + <em>d</em> = <em>R</em>

==>  <em>R</em> = -2 + <em>b</em> + <em>c</em> + <em>d</em>

<em>p</em>(2) = 2(2)³ + <em>b</em>(2)² - <em>c</em>(2) + <em>d</em> = <em>R</em>

==>  <em>R</em> = 16 + 4<em>b</em> - 2<em>c</em> + <em>d</em>

<em>p</em>(1/2) = 2(1/2)³ + <em>b</em>(1/2)² - <em>c</em>(1/2) + <em>d</em> = <em>R</em>

==>  <em>R</em> = 1/4 + <em>b</em>/4 - <em>c</em>/2 + <em>d</em>

<em />

We're also given that <em>y</em> + 2 is a factor, which means dividing <em>p(y)</em> by it leaves no remainder, and so

<em>p</em>(-2) = 2(-2)³ + <em>b</em>(-2)² - <em>c</em>(-2) + <em>d</em> = 0

==>  0 = -16 + 4<em>b</em> + 2<em>c</em> + <em>d</em>

<em />

Solve the system of equations in boldface. You can eliminate <em>d</em> from the first 3 to first solve for <em>b</em> and <em>c</em>, then solve for <em>d</em> :

(-2 + <em>b</em> + <em>c</em> + <em>d</em>) - (16 + 4<em>b</em> - 2<em>c</em> + <em>d</em>) = <em>R</em> - <em>R</em>

-18 - 3<em>b</em> + 3<em>c</em> = 0

<em>b</em> - <em>c</em> = -6

(-2 + <em>b</em> + <em>c</em> + <em>d</em>) - (1/4 + <em>b</em>/4 - <em>c</em>/2 + <em>d</em>) = <em>R</em> - <em>R</em>

-9/4 + 3<em>b</em>/4 + 3<em>c</em>/2 = 0

<em>b</em> + 2<em>c</em> = 3

(<em>b</em> - <em>c</em>) - (<em>b</em> + 2<em>c</em>) = -6 - 3

-3<em>c</em> = -9

<em>c</em> = 3

<em>b</em> - 3 = -6

<em>b</em> = -3

-16 + 4(-3) + 2(3) + <em>d</em> = 0

<em>d</em> = 22

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Use Cramer’s rule to solve for x: x + 4y − z = −14 5x + 6y + 3z = 4 −2x + 7y + 2z = −17
V125BC [204]

Looks like the system is

x + 4y - z = -14

5x + 6y + 3z = 4

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or in matrix form,

\mathbf{Ax} = \mathbf b \iff \begin{bmatrix} 1 & 4 & -1 \\ 5 & 6 & 3 \\ -2 & 7 & 2 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} -14 \\ 4 \\ -17 \end{bmatrix}

Cramer's rule says that

x_i = \dfrac{\det \mathbf A_i}{\det \mathbf A}

where x_i is the solution for i-th variable, and \mathbf A_i is a modified version of \mathbf A with its i-th column replaced by \mathbf b.

We have 4 determinants to compute. I'll show the work for det(A) using a cofactor expansion along the first row.

\det \mathbf A = \begin{vmatrix} 1 & 4 & -1 \\ 5 & 6 & 3 \\ -2 & 7 & 2 \end{vmatrix}

\det \mathbf A = \begin{vmatrix} 6 & 3 \\ 7 & 2 \end{vmatrix} - 4 \begin{vmatrix} 5 & 3 \\ -2 & 2 \end{vmatrix} - \begin{vmatrix} 5 & 6 \\ -2 & 7 \end{vmatrix}

\det \mathbf A = ((6\times2)-(3\times7)) - 4((5\times2)-(3\times(-2)) - ((5\times7)-(6\times(-2)))

\det\mathbf A = 12 - 21 - 40 - 24 - 35 - 12 = -120

The modified matrices and their determinants are

\mathbf A_1 = \begin{bmatrix} -14 & 4 & -1 \\ 4 & 6 & 3 \\ -17 & 7 & 2\end{bmatrix} \implies \det\mathbf A_1 = -240

\mathbf A_2 = \begin{bmatrix} 1 & -14 & -1 \\ 5 & 4 & 3 \\ -2 & -17 & 2 \end{bmatrix} \implies \det\mathbf A_2 = 360

\mathbf A_3 = \begin{bmatrix} 1 & 4 & -14 \\ 5 & 6 & 4 \\ -2 & 7 & -17 \end{bmatrix} \implies \det\mathbf A_3 = -480

Then by Cramer's rule, the solution to the system is

x = \dfrac{-240}{-120} \implies \boxed{x = 2}

y = \dfrac{360}{-120} \implies \boxed{y = -3}

z = \dfrac{-480}{-120} \implies \boxed{z = 4}

5 0
2 years ago
A college graduate expects to earn a salary of 60000 during the first year after graduation and receive a 4% raise every year af
guapka [62]

So, there is an exponential growth formula you use.

First, write the initial cost which is

60,000

Then write 1+ the percentage of growth rate which is 0.04 percent. Then, right the time as an exponent after certain amount of years.

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Plug this in a calculator

it should be 88814.6571

I hope this helps.

6 0
3 years ago
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