Factor the high-degree polynomial 50x5−32x to solve the polynomial equation 50x5−32x=0 .(1 point)

Question

Responses

x=2/5√, x=−2/5√
x equals Start Fraction 2 over Start Root 5 End Root End Fraction , x equals negative Start Fraction 2 over Start Root 5 End Root End Fraction

x=4/5, x=−4/5
x equals Start Fraction 4 over 5 End Fraction , x equals negative Start Fraction 4 over 5 End Fraction

x=2/5√
x equals Start Fraction 2 over Start Root 5 End Root End Fraction

x=0, x=2/5√, x=−2/5√

Answer 1

To factor the polynomial 50x^5 - 32x, we can first factor out x:
x(50x^4 - 32)

Next, we can factor out 2 from the expression inside the parentheses:
x(2(25x^4 - 16))

Now, we can factor a difference of squares:
x(2(5x^2 - 4)(5x^2 + 4))

This gives us the factored form of the polynomial as:
x(2)(5x^2 - 4)(5x^2 + 4)

To solve the polynomial equation 50x^5 - 32x = 0, we can set each factor equal to zero:

x = 0
5x^2 - 4 = 0
5x^2 + 4 = 0

Solving for x in each equation, we get:

x = 0

For 5x^2 - 4 = 0:
5x^2 = 4
x^2 = 4/5
x = ±√(4/5)
x = ±(2/√5)
x = ±(2/5√5)

For 5x^2 + 4 = 0:
5x^2 = -4
x^2 = -4/5
There are no real solutions for this equation since the square of a real number cannot be negative.

Therefore, the solutions to the polynomial equation 50x^5 - 32x = 0 are:

x = 0, x = 2/5√5, x = -2/5√5