Find the zeros of f(x) = x^3 +9x^2 + 23x + 15

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hala718 | High School Teacher | (Level 1) Educator Emeritus

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f(x) = x^3 +9x^2 + 23x + 15

To solve, we will subsitute with the 15 factors which are:

1, -1, 3, -3, 5, -5, 15, -15

let us subsitute with x= -5:

f(-5) = 5^3 + 9*5^2 + 23*5 + 15

      = -125 + 225 - 529 + 15 = 0

The (x+5) is one of the factors:

==> f(x) = (x+ 5) * g(x)

Now we will divide f(x) by (x+5):

==> f(x) = (x+ 5) (x^2 + 4x + 3)

Now factor (x^2 + 4x + 3)

==> f(x) = (x+5)(x+3)(x+1)

Now we will determine the zeros:

==> x1= -5

==> x2= -3

==> x3= -1

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neela | High School Teacher | (Level 3) Valedictorian

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To find the zeros of  f(x) = x^3 +9x^2 + 23x + 15.

We observe that if we put x= -1, then f(-1) = (-1)+9-23+15 = 0.

So x+1 is a facor of the f(x).

Therefore x^3+9x^2+23x+15 = (x+1))(x^2 +kx+15) , as this agrees with x^3 and constant terms.

Now equating x terms on both sides, 23x = 15x+kx.

kx = 23x-15x

kx = 8x.

k = 8.

Therefore x^2+8x+15 is a factor.

x^2+8x+15 = (x+5)(x+3).

Therefore  f(x) = x^3+9x^2+15x+23 = (x+1)(x+3)(x+5).

Therefore x+1 = 0 , x+5 = 0 , x+3 = 0 gives the zeros of f(x)

Or x = -1, x= -5 , or x= -3 are the zeros of f(x).

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