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The binomial [tex][tex]$8 x^3 + 125 y^3$[/tex][/tex] can be factored as [tex][tex]$(2 x + 5 y)\left(4 x^2 - ? + 25 y^2\right)$[/tex][/tex].

Find the missing term.
(Do not include the minus sign.)

The solution is [tex]\square[/tex]



Answer :

To solve the problem of factoring the binomial [tex]\(8x^3 + 125y^3\)[/tex], we need to use the sum of cubes formula. The sum of cubes formula is expressed as:

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

First, we identify the terms [tex]\(a\)[/tex] and [tex]\(b\)[/tex] from the given binomial [tex]\(8x^3 + 125y^3\)[/tex]:

[tex]\[ a^3 = 8x^3 \quad \text{and} \quad b^3 = 125y^3 \][/tex]

Since [tex]\(8x^3 = (2x)^3\)[/tex] and [tex]\(125y^3 = (5y)^3\)[/tex], we can identify:

[tex]\[ a = 2x \quad \text{and} \quad b = 5y \][/tex]

Next, applying the sum of cubes formula [tex]\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)[/tex]:

[tex]\[ 8x^3 + 125y^3 = (2x + 5y)((2x)^2 - (2x)(5y) + (5y)^2) \][/tex]

Now we expand and simplify the expression inside the parentheses:

[tex]\[ (2x + 5y)((2x)^2 - (2x)(5y) + (5y)^2) \][/tex]

Calculate each term individually:

[tex]\[ (2x)^2 = 4x^2 \][/tex]
[tex]\[ (2x)(5y) = 10xy \][/tex]
[tex]\[ (5y)^2 = 25y^2 \][/tex]

Putting it all together in the expression:

[tex]\[ (2x + 5y)(4x^2 - 10xy + 25y^2) \][/tex]

From this, we can see that the missing term inside the parentheses is [tex]\(10xy\)[/tex].

Therefore, the missing term in the factorization [tex]\(8x^3 + 125y^3 = (2x + 5y)(4x^2 - ? + 25y^2)\)[/tex] is:

[tex]\[ 10xy \][/tex]

So, the answer is:

[tex]\[ \boxed{10xy} \][/tex]