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3 years ago

Find m and n if the remainder when 8x^3+mx^2-6x+n if divided by (x-1) and (2x-3) are 2 and 8 respectively

Mathematics

1 answer:

soldier1979 [14.2K]3 years ago

3 0

Answer:

we have:

8x³ + mx² - 6x + n

= 8x³ - 8x² + (m + 8)x²- (m + 8)x + (m + 2)x - (m + 2) + m + 2+ n

= 8x²(x - 1) + (m + 8)x(x - 1) + (m + 2)(x - 1) + (m + n + 2)

= (x - 1)[8x² + (m + 8)x + m + 2] + (m + n + 2)

because the remainder if divided by (x-1) is 2

=> m + n + 2 = 2

⇔ m + n = 0 (1)

we also have:

8x³ + mx² - 6x + n

= 8x³ - 12x² + (m + 12)x² - 3/2.x.(m + 12) + ( 12 + 3/2.m)x - (9/4.m + 18) + n +9/4m + 18

= 4x²(2x - 3) + 1/2.(m + 12)x(2x - 3) + (3/2m + 12).1/2.(2x - 3) + 9/4m + n + 18

= (2x - 3)(4x² + (m + 12)/2.x + 3/4m + 6) + 9/4m + n + 18

because the remainder if divided by (2x - 3) is 8

=> 9/4m + n + 18 = 8

⇔ 9/4m + n = -10 (2)

from (1) and (2), we have:

m + n = 0

9/4m + n = -10

=> m = -8

n = 8

Step-by-step explanation:

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pav-90 [236]

we know that

if the exponential function passes through the given point, then the point must satisfy the equation of the exponential function

we proceed to verify each case if the point $(2,80)$ satisfied the exponential function

case A $f(x)=4(x^{5})$

For $x=2$ calculate the value of y in the equation and then compare with the y-coordinate of the point

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$128\neq 80$

therefore

the exponential function $f(x)=4(x^{5})$ not passes through the point $(2,80)$

case B $f(x)=5(x^{4})$

For $x=2$ calculate the value of y in the equation and then compare with the y-coordinate of the point

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$80=80$

therefore

the exponential function $f(x)=5(x^{4})$ passes through the point $(2,80)$

case C $f(x)=4(5^{x})$

For $x=2$ calculate the value of y in the equation and then compare with the y-coordinate of the point

$f(2)=4(5^{2})=100$

$100\neq 80$

therefore

the exponential function $f(x)=4(5^{x})$ not passes through the point $(2,80)$

case D $f(x)=5(4^{x})$

For $x=2$ calculate the value of y in the equation and then compare with the y-coordinate of the point

$f(2)=5(4^{2})=80$

$80=80$

therefore

the exponential function $f(x)=5(4^{x})$ passes through the point $(2,80)$

therefore

the answer is

$f(x)=5(x^{4})$

$f(x)=5(4^{x})$

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