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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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