Solve the polynomial equation by factoring
2x^3 - 5x^2 - 3x = 0
Baby steps please...I believe answer is
X= 0, 3, -1/2
But I don't know why
Thank you!
first take out a common factor of x
x(2x^2 - 5x - 3) = 0
x(x-3)(2x+1) = 0 , I assume you know how to factor quadratic trinomials
set each factor equal to zero
x = 0
x-3 = 0 ---> x = 3
2x+1 = 0 ----> x = -1/2
To solve the given polynomial equation by factoring, we need to factor out the common terms and then use the factored form to find the values of x that make the equation equal to zero.
Step 1: Look for the common factor
In this equation, we can see that x is a common factor for all the terms, so we can factor it out:
x(2x^2 - 5x - 3) = 0
Step 2: Factor the trinomial expression
Now, we need to factor the remaining trinomial expression: 2x^2 - 5x - 3.
To do this, we look for two numbers that multiply to give -6 (the product of the leading coefficient and the constant term) and add up to -5 (the coefficient of the x term).
The two numbers that fit these conditions are -6 and +1.
So, we can rewrite the trinomial as follows:
2x^2 - 6x + x - 3 = 0
Step 3: Group terms and factor by grouping
Next, we group the terms in pairs:
(x is common in the first pair, and -3 is common in the second pair)
x(2x - 6) + 1(2x - 6) = 0
Now, we factor out the common factor from each pair:
x(2x - 6) + 1(2x - 6) = (2x - 6)(x + 1) = 0
Step 4: Apply the zero product property
According to the zero product property, if the product of two factors is equal to zero, then at least one of the factors must be zero. So, set each factor equal to zero and solve for x:
2x - 6 = 0 --> 2x = 6 --> x = 3
x + 1 = 0 --> x = -1
Therefore, the solutions to the equation 2x^3 - 5x^2 - 3x = 0 are x = 0, x = 3, and x = -1/2.