Get the information you need quickly and easily with IDNLearn.com. Join our interactive Q&A community and get reliable, detailed answers from experienced professionals across a variety of topics.
Sagot :
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]
[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]
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. Discover the answers you need at IDNLearn.com. Thanks for visiting, and come back soon for more valuable insights.