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The expression \( 4x^{4} - 12x^{3} + 8x^{2} \) can be factored as \( 4x^{2}(x - 1)(x - 2) \).
Solution
Let's factor the expression 4x⁴ - 12x³ + 8x² step by step.
Step 1: Factor out the greatest common factor (GCF).
Each term has a factor of 4x², so we factor that out:
4x⁴ - 12x³ + 8x² = 4x²(x² - 3x + 2)
Step 2: Factor the quadratic inside the parentheses.
We have the quadratic x² - 3x + 2. Look for two numbers that multiply to 2 and add to -3.
These numbers are -1 and -2 because:
(-1) × (-2) = 2
(-1) + (-2) = -3
Thus, we can factor the quadratic as:
x² - 3x + 2 = (x - 1)(x - 2)
Step 3: Write the fully factored form.
Substitute the factored quadratic back in:
4x⁴ - 12x³ + 8x² = 4x² (x - 1)(x - 2)
This is the final factored form of the expression.
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Did you know that polynomials like \(4x^4 - 12x^3 + 8x^2\) have a wide range of applications in physics, economics, and even computer graphics? They can represent everything from the trajectory of a projectile to modeling population growth, making them a crucial tool in diverse fields! When factoring a polynomial, a common mistake is to overlook the greatest common factor (GCF). In this case, you can first factor out \(4x^2\) from the polynomial, simplifying the process and making it easier to solve or analyze. So remember, always start by checking for a GCF!
