What is the square root of 4x 4 + 8x 3 - 4x + 1?
2x 2+ 2x - 1
Polynomial Square Root Calculation
The question asks for the square root of the given polynomial: \(4x^4 + 8x^3 - 4x + 1\). We need to find which of the provided options, when squared, equals this polynomial.
Polynomial Square Root Verification Steps
To find the correct square root, we can test each option by squaring it. The option whose square matches the original polynomial \(4x^4 + 8x^3 - 4x + 1\) is the correct answer. Let's check the options provided.
Checking Option 4: \(2x^2 + 2x - 1\)
We will test the fourth option, \(2x^2 + 2x - 1\). We need to calculate its square:
\[ (2x^2 + 2x - 1)^2 \]
Let's expand this expression by multiplying it by itself:
\[ (2x^2 + 2x - 1) \times (2x^2 + 2x - 1) \]
We use the distributive property to multiply the terms:
- Multiplying \(2x^2\) by \((2x^2 + 2x - 1)\): \[ 2x^2 \times (2x^2 + 2x - 1) = 4x^4 + 4x^3 - 2x^2 \]
- Multiplying \(2x\) by \((2x^2 + 2x - 1)\): \[ 2x \times (2x^2 + 2x - 1) = 4x^3 + 4x^2 - 2x \]
- Multiplying \(-1\) by \((2x^2 + 2x - 1)\): \[ -1 \times (2x^2 + 2x - 1) = -2x^2 - 2x + 1 \]
Now, we add the results of these multiplications:
\[ (4x^4 + 4x^3 - 2x^2) + (4x^3 + 4x^2 - 2x) + (-2x^2 - 2x + 1) \]
Combine the like terms:
- Terms with \(x^4\): \(4x^4\)
- Terms with \(x^3\): \(4x^3 + 4x^3 = 8x^3\)
- Terms with \(x^2\): \(-2x^2 + 4x^2 - 2x^2 = 0x^2\)
- Terms with \(x\): \(-2x - 2x = -4x\)
- Constant terms: \(1\)
Summing these combined terms, we get:
\[ 4x^4 + 8x^3 + 0x^2 - 4x + 1 \]
This expression simplifies to:
\[ 4x^4 + 8x^3 - 4x + 1 \]
The result of squaring \(2x^2 + 2x - 1\) matches the original polynomial given in the question.
Final Answer Determination
Since squaring the expression \(2x^2 + 2x - 1\) results in the polynomial \(4x^4 + 8x^3 - 4x + 1\), this expression is the correct square root of the polynomial.
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