Question

Four consective natural numbers have product 840.Find numbers.​

Answer 1

$\huge{\underline{\underline{\green{\mathfrak{Answer:}}}}}$

let the four consecutive numbers be x, (x+1), (x+2) and (x+3) respectively

According to the question:

x (x+1) (x+2) (x+3) = 840

=> (x² + x) (x² + 5x + 3) = 840

=> x⁴ + 6x³ + 11x² + 6x = 840

Now putting values for this polynomial, p(x)

p(x) = x⁴ + 6x³ + 11x² + 6x = 840

=> p(x) = x⁴ + 6x³ + 11x² + 6x - 840

If x = 1

p(1) = (1)⁴ + 6(1)³ + 11(1)² + 6(1) - 840

= -816

If x = 2

p(2) = (2)⁴ + 6(2)³ + 11(2)² + 6(2) - 840

= -720

If x = 3

p(3) = (3)⁴ + 6(3)³ + 11(3)² + 6(3) - 840

= -480

If x = 4

p(4) = (4)⁴ + 6(4)³ + 11(4)² + 6(4) - 840

= 0

So, x = 4 (First number)

(x+1) = 5 (Second number)

(x+2) = 6 (Third number)

(x+3) = 7 (Fourth number)

So, the consecutive numbers are 4,5,6 and 7

Answer 2

Answer:

let the four consecutive numbers be x, (x+1), (x+2) and (x+3) respectively

According to the question:

x (x+1) (x+2) (x+3) = 840

=> (x² + x) (x² + 5x + 3) = 840

=> x⁴ + 6x³ + 11x² + 6x = 840

Now putting values for this polynomial, p(x)

p(x) = x⁴ + 6x³ + 11x² + 6x = 840

=> p(x) = x⁴ + 6x³ + 11x² + 6x - 840

If x = 1

p(1) = (1)⁴ + 6(1)³ + 11(1)² + 6(1) - 840

= -816

If x = 2

p(2) = (2)⁴ + 6(2)³ + 11(2)² + 6(2) - 840

= -720

If x = 3

p(3) = (3)⁴ + 6(3)³ + 11(3)² + 6(3) - 840

= -480

If x = 4

p(4) = (4)⁴ + 6(4)³ + 11(4)² + 6(4) - 840

= 0

So, x = 4 (First number)

(x+1) = 5 (Second number)

(x+2) = 6 (Third number)

(x+3) = 7 (Fourth number)

So, the consecutive numbers are 4,5,6 and 7