Greg is solving a polynomial equation and gets stuck at this step: x^2(2x−1)+5x(2x−1)+4(2x−1) Which of the following is a solution? (Select all that apply) (3 points)
1. -1
2. -4
3. 1
4. 1/2
5. 0
To find the solutions to the given polynomial equation, we need to factor the expression and set it equal to zero.
The given equation is: x^2(2x−1) + 5x(2x−1) + 4(2x−1)
Let's factor out the common term (2x−1):
(2x−1)(x^2 + 5x + 4)
Now, we can set each factor equal to zero and solve for x:
1. Setting 2x − 1 = 0:
2x = 1
x = 1/2
2. Factoring the quadratic equation x^2 + 5x + 4 and setting it equal to zero:
(x + 1)(x + 4) = 0
Setting (x + 1) = 0:
x = -1
Setting (x + 4) = 0:
x = -4
Therefore, the solutions to the given polynomial equation are:
1. x = -1
2. x = -4
3. x = 1/2
So, the correct answers are:
1. -1
2. -4
4. 1/2
To find the solution(s) to the polynomial equation, we need to simplify the expression and then solve for x. Let's simplify the given expression step by step:
x^2(2x−1) + 5x(2x−1) + 4(2x−1)
First, let's distribute the terms inside the parentheses:
2x^3 - x^2 + 10x^2 - 5x + 8x - 4
Next, combine like terms:
2x^3 + 9x^2 + 3x - 4
Now that we have simplified the expression, we can set it equal to zero to find the solutions. We have:
2x^3 + 9x^2 + 3x - 4 = 0
Since we are given options to choose from, we can substitute each value into the equation and check if it satisfies the equation.
Option 1: Substituting x = -1 into the equation:
2(-1)^3 + 9(-1)^2 + 3(-1) - 4 = -2 + 9 - 3 - 4 = 0
So, -1 is a solution to the equation.
Option 2: Substituting x = -4 into the equation:
2(-4)^3 + 9(-4)^2 + 3(-4) - 4 = -128 + 144 - 12 - 4 = 0
So, -4 is a solution to the equation.
Option 3: Substituting x = 1 into the equation:
2(1)^3 + 9(1)^2 + 3(1) - 4 = 2 + 9 + 3 - 4 = 10
So, 1 is not a solution to the equation.
Option 4: Substituting x = 1/2 into the equation:
2(1/2)^3 + 9(1/2)^2 + 3(1/2) - 4 = 1/4 + 9/4 + 3/2 - 4 = 16/4 - 4 = 0
So, 1/2 is a solution to the equation.
Option 5: Substituting x = 0 into the equation:
2(0)^3 + 9(0)^2 + 3(0) - 4 = -4 ≠ 0
So, 0 is not a solution to the equation.
Therefore, the solutions to the equation are:
1. -1
2. -4
4. 1/2
So, options 1, 2, and 4 are solutions to the given polynomial equation.
To find the solutions of the polynomial equation, we can factor out the common factor of (2x-1):
x^2(2x-1) + 5x(2x-1) + 4(2x-1)
= (2x-1)(x^2 + 5x + 4)
Now, we need to find the values of x that make the expression (2x-1)(x^2 + 5x + 4) equal to zero. Thus, the solutions are the values of x that make either (2x-1) or (x^2 + 5x + 4) equal to zero.
To solve for (2x-1) = 0, we get:
2x-1 = 0
2x = 1
x = 1/2
To solve for (x^2 + 5x + 4) = 0, we can factor the quadratic expression:
(x+4)(x+1) = 0
x+4 = 0 or x+1 = 0
x = -4 or x = -1
Therefore, the solutions to the polynomial equation are:
1. x = 1/2
2. x = -4
3. x = -1
So, the correct answers are 1, 2, and 3.