Approximate the real number solution(s) to the polynomial function f(x) = x3 + 2x2 - 5x - 6.

Solution:

" A polynomial is a type of expression in which the exponents of all variables should be a whole number." ​​​​​​​

Given, f(x) = x3 + 2x2 - 5x - 6.

By trial and error method,

Let x = 1

f(1) = 13 + 2(1)2 - 5(1) - 6

= 1 + 2 - 5 - 6

= -7

f(1) ≠ 0

Now, consider x = -1

f(-1) = (-1)3 + 2(-1)2 - 5(-1) - 6

= -1 + 2 + 5 - 6

= 0

f(-1) = 0

(x+1) is a factor of f(x) = x3 + 2x2 - 5x - 6.

Let x = 2

f(2) = (2)3 + 2(2)2 - 5(2) - 6

= 8 + 8 - 10 - 6

= 16 - 16

= 0

f(2) = 0

(x - 2) is a factor of f(x) = x3 + 2x2 - 5x - 6.

Let x = -2

f(-2) = (-2)3 + 2(-2)2 - 5(-2) - 6

= -8 + 8 + 10 - 6

= 4

f(-2) ≠ 0

Let x = 3

f(3) = (3)3 + 2(3)2 - 5(3) - 6

= 27 + 18 - 15 - 6

= 24

f(3) ≠ 0

Let x = -3

f(-3) = (-3)3 + 2(-3)2 - 5(-3) - 6

= -27 + 18 + 15 - 6

= -33 + 33

=0

f(-3) = 0

(x + 3) is a factor of f(x) = x3 + 2x2 - 5x - 6.

Therefore, the solutions are x = -1, 2 and -3.

Approximate the real number solution(s) to the polynomial function f(x) = x3 + 2x2 - 5x - 6.

Summary:

The real number solution(s) to the polynomial function f(x) = x3 + 2x2 - 5x - 6 are -1, 2 and -3.

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