Question

Using factoring by grouping, find all of the zeros (solutions) of the given equation

(Hint: There should be 3 real solutions) x^3+ 5x^2-2x-10=0

Answer 1

To use factoring by grouping to solve the equation x^3 + 5x^2 - 2x - 10 = 0, we can group the first two terms and the last two terms:

x^2(x + 5) - 2(x + 5) = 0

Now we can factor out the common factor of (x + 5):

(x + 5)(x^2 - 2) = 0

This gives us two possible solutions:

x + 5 = 0, which gives us x = -5

or

x^2 - 2 = 0, which gives us x = ±√2

Therefore, the solutions to the equation x^3 + 5x^2 - 2x - 10 = 0 are x = -5, x = √2, and x = -√2. All three solutions are real

Answer 2

x = - 5 x = ±
`√(2)`

Explanation:

x³ + 5x² - 2x - 10 = 0 ( factor the first/second and third/fourth terms )

x²(x + 5) - 2(x + 5) = 0 ← factor out (x + 5) from each term

(x + 5)(x² - 2) = 0

equate each factor to zero and solve for x

x + 5 = 0 ⇒ x = - 5

x² - 2 = 0 ( add 2 to both sides )

x² = 2 ( take square root of both sides )

x = ±
`√(2)`

then the solutions are

x = - 5 , x = -
`√(2)` , x =
`√(2)`