Answer :
Let's solve the polynomial equation [tex]\( 4x^4 - 36x^2 = 0 \)[/tex] by factoring and then verify the solutions by substitution.
### Step 1: Set the Equation to Zero
We start with the polynomial equation:
[tex]\[ 4x^4 - 36x^2 = 0 \][/tex]
### Step 2: Factor Out the Greatest Common Factor
First, we look for the greatest common factor (GCF) that can be factored out. Both terms have a common factor of [tex]\( 4x^2 \)[/tex]:
[tex]\[ 4x^4 - 36x^2 = 4x^2(x^2 - 9) \][/tex]
### Step 3: Further Factorization
Next, we notice that the expression [tex]\( x^2 - 9 \)[/tex] is a difference of squares, which can be factored as follows:
[tex]\[ x^2 - 9 = (x - 3)(x + 3) \][/tex]
Therefore, our factored equation becomes:
[tex]\[ 4x^2(x - 3)(x + 3) = 0 \][/tex]
### Step 4: Set Each Factor to Zero
To find the solutions, we set each factor equal to zero:
[tex]\[ 4x^2 = 0 \quad \text{or} \quad x - 3 = 0 \quad \text{or} \quad x + 3 = 0 \][/tex]
### Step 5: Solve Each Equation
- For [tex]\( 4x^2 = 0 \)[/tex]:
[tex]\[ x^2 = 0 \quad \Rightarrow \quad x = 0 \][/tex]
- For [tex]\( x - 3 = 0 \)[/tex]:
[tex]\[ x = 3 \][/tex]
- For [tex]\( x + 3 = 0 \)[/tex]:
[tex]\[ x = -3 \][/tex]
So the solutions to the equation [tex]\( 4x^4 - 36x^2 = 0 \)[/tex] are:
[tex]\[ x = 0, 3, -3 \][/tex]
### Step 6: Check by Substitution
We will verify each solution by substituting it back into the original equation [tex]\( 4x^4 - 36x^2 = 0 \)[/tex]:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ 4(0)^4 - 36(0)^2 = 0 - 0 = 0 \quad \text{(True)} \][/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ 4(3)^4 - 36(3)^2 = 4(81) - 36(9) = 324 - 324 = 0 \quad \text{(True)} \][/tex]
- For [tex]\( x = -3 \)[/tex]:
[tex]\[ 4(-3)^4 - 36(-3)^2 = 4(81) - 36(9) = 324 - 324 = 0 \quad \text{(True)} \][/tex]
Each substitution checks out, confirming that the solutions [tex]\( x = 0, 3, \text{ and } -3 \)[/tex] are correct.
### Summary
We have factored and solved the polynomial equation [tex]\( 4x^4 - 36x^2 = 0 \)[/tex], and verified the solutions by substitution. The solutions are:
[tex]\[ x = 0, 3, -3 \][/tex]
### Step 1: Set the Equation to Zero
We start with the polynomial equation:
[tex]\[ 4x^4 - 36x^2 = 0 \][/tex]
### Step 2: Factor Out the Greatest Common Factor
First, we look for the greatest common factor (GCF) that can be factored out. Both terms have a common factor of [tex]\( 4x^2 \)[/tex]:
[tex]\[ 4x^4 - 36x^2 = 4x^2(x^2 - 9) \][/tex]
### Step 3: Further Factorization
Next, we notice that the expression [tex]\( x^2 - 9 \)[/tex] is a difference of squares, which can be factored as follows:
[tex]\[ x^2 - 9 = (x - 3)(x + 3) \][/tex]
Therefore, our factored equation becomes:
[tex]\[ 4x^2(x - 3)(x + 3) = 0 \][/tex]
### Step 4: Set Each Factor to Zero
To find the solutions, we set each factor equal to zero:
[tex]\[ 4x^2 = 0 \quad \text{or} \quad x - 3 = 0 \quad \text{or} \quad x + 3 = 0 \][/tex]
### Step 5: Solve Each Equation
- For [tex]\( 4x^2 = 0 \)[/tex]:
[tex]\[ x^2 = 0 \quad \Rightarrow \quad x = 0 \][/tex]
- For [tex]\( x - 3 = 0 \)[/tex]:
[tex]\[ x = 3 \][/tex]
- For [tex]\( x + 3 = 0 \)[/tex]:
[tex]\[ x = -3 \][/tex]
So the solutions to the equation [tex]\( 4x^4 - 36x^2 = 0 \)[/tex] are:
[tex]\[ x = 0, 3, -3 \][/tex]
### Step 6: Check by Substitution
We will verify each solution by substituting it back into the original equation [tex]\( 4x^4 - 36x^2 = 0 \)[/tex]:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ 4(0)^4 - 36(0)^2 = 0 - 0 = 0 \quad \text{(True)} \][/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ 4(3)^4 - 36(3)^2 = 4(81) - 36(9) = 324 - 324 = 0 \quad \text{(True)} \][/tex]
- For [tex]\( x = -3 \)[/tex]:
[tex]\[ 4(-3)^4 - 36(-3)^2 = 4(81) - 36(9) = 324 - 324 = 0 \quad \text{(True)} \][/tex]
Each substitution checks out, confirming that the solutions [tex]\( x = 0, 3, \text{ and } -3 \)[/tex] are correct.
### Summary
We have factored and solved the polynomial equation [tex]\( 4x^4 - 36x^2 = 0 \)[/tex], and verified the solutions by substitution. The solutions are:
[tex]\[ x = 0, 3, -3 \][/tex]
