Get comprehensive solutions to your questions with the help of IDNLearn.com's experts. Get accurate answers to your questions from our community of experts who are always ready to provide timely and relevant solutions.
Sagot :
To find the product of the polynomials \((a + b + c)(a² + b² + c² + ab - bc - ca)\), we can use the distributive property (also known as the FOIL method for binomials). Here's the step-by-step process:
1. Distribute each term in the first polynomial \((a + b + c)\) to each term in the second polynomial \((a² + b² + c² + ab - bc - ca)\).
Let's break it down:
\[
(a + b + c)(a² + b² + c² + ab - bc - ca)
\]
First, distribute \(a\):
\[
a(a² + b² + c² + ab - bc - ca) = a³ + ab² + ac² + a²b - abc - a²c
\]
Next, distribute \(b\):
\[
b(a² + b² + c² + ab - bc - ca) = ba² + b³ + bc² + ab² - b²c - bac
\]
Finally, distribute \(c\):
\[
c(a² + b² + c² + ab - bc - ca) = ca² + cb² + c³ + cab - cbc - c²a
\]
Now, combine all the terms:
\[
a³ + ab² + ac² + a²b - abc - a²c + ba² + b³ + bc² + ab² - b²c - bac + ca² + cb² + c³ + cab - cbc - c²a
\]
Combine like terms:
\[
a³ + b³ + c³ + 2ab² + 2ac² + 2a²b + 2bc² + 2cab - abc - b²c - cbc - a²c - bac - c²a
\]
This is the expanded form of the product of the given polynomials.
1. Distribute each term in the first polynomial \((a + b + c)\) to each term in the second polynomial \((a² + b² + c² + ab - bc - ca)\).
Let's break it down:
\[
(a + b + c)(a² + b² + c² + ab - bc - ca)
\]
First, distribute \(a\):
\[
a(a² + b² + c² + ab - bc - ca) = a³ + ab² + ac² + a²b - abc - a²c
\]
Next, distribute \(b\):
\[
b(a² + b² + c² + ab - bc - ca) = ba² + b³ + bc² + ab² - b²c - bac
\]
Finally, distribute \(c\):
\[
c(a² + b² + c² + ab - bc - ca) = ca² + cb² + c³ + cab - cbc - c²a
\]
Now, combine all the terms:
\[
a³ + ab² + ac² + a²b - abc - a²c + ba² + b³ + bc² + ab² - b²c - bac + ca² + cb² + c³ + cab - cbc - c²a
\]
Combine like terms:
\[
a³ + b³ + c³ + 2ab² + 2ac² + 2a²b + 2bc² + 2cab - abc - b²c - cbc - a²c - bac - c²a
\]
This is the expanded form of the product of the given polynomials.
Answer:
a³ + b³ + c³ - abc + 2ab(a + b)
Step-by-step explanation:
(a + b + c)(a² + b² + c² + ab - bc - ca)
= a³ + ab² + ac² + a²b - abc - a²c +
a²b + b³ + bc² + ab² - b²c - abc +
a²c + b²c + c³ + abc - bc² - ac²
= a³ + b³ + c³ + 2ab² + 2a²b - abc
= a³ + b³ + c³ - abc + 2ab(a + b)
We are happy to have you as part of our community. Keep asking, answering, and sharing your insights. Together, we can create a valuable knowledge resource. IDNLearn.com has the solutions you’re looking for. Thanks for visiting, and see you next time for more reliable information.
