Question

the values of l and m so that lx4+mx3+2x2+4 is exectally divisible by x2-x-2 is rescepectively

Answer 1

Solution :

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Given :

$lx^4 + mx^3 + 2x^2 + 4$

is exactly divisible by x² - x - 2

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To Find :

The values of l & m.

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As we know that,

the equation :

$lx^4 + mx^3 + 2x^2 + 4$ is exactly divisible by x² - x - 2,.

Hence the factors of x² - x - 2 are divisible by $lx^4 + mx^3 + 2x^2 + 4$

Hence,

⇒ x² - x - 2

⇒ x² - 2x + x - 2

⇒ x(x - 2) + (x - 2)

⇒ (x + 1)(x - 2)

if the value of this equation is 0,.

then,

x + 1 = 0 (or) x - 2 = 0

⇒ x = -1 (or) x = 2

So,


P(2) = 0 , P(-1) = 0,. (As, we know)

if,

P(x) = $lx^4 + mx^3 + 2x^2 + 4$

⇒ $P(2) = l(2)^4 + m(2)^3 + 2(2)^2 + 4$

⇒ 0 = $l(16) + m(8) + 2(4)+ 4$

⇒ $0 = 16l + 8m + 8 + 4$

⇒ $0 = 16l + 8m + 12$

⇒ $16l + 8m = - 12$ ....(i)

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P(x) = $lx^4 + mx^3 + 2x^2 + 4$

⇒ P(-1) = $l(-1)^4 + m(-1)^3 + 2(-1)^2 + 4$

⇒ 0 = $l(1) + m(-1) + 2(1) + 4$

⇒ 0 = l - m + 2 + 4

⇒ 0 = l - m + 6

⇒ l - m = -6 .....(ii)


(i) + (ii) x 8 ⇒ 16l + 8m + (l - m)8 = -12 + (-6)(8)

⇒ 16l + 8m + 8l - 8m = -12 - 48

⇒ 24l = - 60

⇒ l = -2.5

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(ii) ⇒ l - m = -6

⇒ - 2.5 - m = -6

⇒ - m = -6 + 2.5

⇒ -m = -3.5

⇒ m = 3.5

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Answer 2

Let f(x)=lx^4+mx^3+2x^2+4  is exactly divisible by

x^2−x−2

f(x)=0

x^2−x−2⇒(x−2)(x+1)

f(2)=l(2)^4+m(2)^3+2(2)^2+4=0

=16l+8m+8+4=0

=16l+8m+12=0

=4(4l+2m+3)=0

=4l+2m+3=0→(1)

f(−1)=l(−1)^4+m(−1)^3+2(−1)^2+4=0

=l−m+2+4=0

=l−m+6=0→(2)

Solve (1) and (2)

l=−15/6

l=−5/2

l=−5/2  is substitute  in equation (2)

−5/2+6/1=m

=−5+12/2=m=>7/2=m