To determine which of the given expressions is a factor of \( 8x^2 - 2 \) through long division, we first factor the expression itself if possible.
We start by factoring \( 8x^2 - 2 \):
\[ 8x^2 - 2 = 2(4x^2 - 1) = 2(2x - 1)(2x + 1) \]
So, our expression \( 8x^2 - 2 \) factors into \( 2(2x - 1)(2x + 1) \).
Now let's check each of the expressions provided:
- \( x + 3 \)
- \( 2x + 1 \)
- \( x + 1 \)
- \( 2x + 3 \)
From our factorization, we know that \( 2x + 1 \) is indeed a factor of \( 8x^2 - 2 \).
We can also perform long division to confirm this. Let's divide \( 8x^2 - 2 \) by \( 2x + 1 \):
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Divide the leading term: \[ \frac{8x^2}{2x} = 4x \]
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Multiply \( 4x \) by \( 2x + 1 \): \[ 4x(2x + 1) = 8x^2 + 4x \]
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Subtract this from \( 8x^2 - 2 \): \[ (8x^2 - 2) - (8x^2 + 4x) = -4x - 2 \]
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Next, divide the leading term \( -4x \) by \( 2x \): \[ \frac{-4x}{2x} = -2 \]
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Multiply \( -2 \) by \( 2x + 1 \): \[ -2(2x + 1) = -4x - 2 \]
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Subtract this from \( -4x - 2 \): \[ (-4x - 2) - (-4x - 2) = 0 \]
The remainder is \( 0 \), confirming that \( 2x + 1 \) is indeed a factor of \( 8x^2 - 2 \).
Thus, the correct answer is:
2x + 1
1 answer