To factor the polynomial \( 9x^{10} - 36x^2 \), we can start by identifying any common factors in both terms.
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Identify the greatest common factor (GCF): The GCF of the coefficients \( 9 \) and \( 36 \) is \( 9 \). Additionally, both terms have \( x^2 \) as a common factor. Thus, we can factor out \( 9x^2 \).
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Factor out \( 9x^2 \): \[ 9x^{10} - 36x^2 = 9x^2(x^8 - 4) \]
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Recognize the difference of squares: The expression \( x^8 - 4 \) is a difference of squares, since \( 4 \) can be written as \( 2^2 \): \[ x^8 - 4 = (x^4)^2 - (2)^2 \] This can be factored using the difference of squares formula, \( a^2 - b^2 = (a - b)(a + b) \): \[ x^8 - 4 = (x^4 - 2)(x^4 + 2) \]
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Combine the factors: Putting it all together, we have: \[ 9x^{10} - 36x^2 = 9x^2(x^4 - 2)(x^4 + 2) \]
Thus, the factored form of the polynomial \( 9x^{10} - 36x^2 \) is: \[ 9x^2(x^4 - 2)(x^4 + 2) \]