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125x³+ 64
Factor the following polynomial

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Final Answer:

The factored form of the polynomial is \[ (5x + 4)(25x^2 - 20x + 16) \]

Step-by-step explanation:

To factor the polynomial \( 125x^3 + 64 \), we can look at the sum of cubes formula. The sum of cubes formula is given by:

\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]

The polynomial \( 125x^3 + 64 \) can be written as a sum of cubes because \( 125 \) is the cube of \( 5 \) and \( 64 \) is the cube of \( 4 \). So we have:

\[ 125x^3 + 64 = (5x)^3 + 4^3 \]

Now we can recognize \( a = 5x \) and \( b = 4 \), and apply the formula:

\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]

Substitute \( a = 5x \) and \( b = 4 \):

\[ (5x)^3 + 4^3 = (5x + 4)((5x)^2 - (5x)(4) + 4^2) \]

Now let's expand and simplify the terms:

\[ = (5x + 4)(25x^2 - 20x + 16) \]

Therefore, the factored form of the given polynomial \( 125x^3 + 64 \) is:

\[ (5x + 4)(25x^2 - 20x + 16) \]

User Fahimeh Ahmadi
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