Calculate the algebraic expression E=(2x+y) (5a-b)^3
To calculate the algebraic expression E=(2x+y) (5a-b)^3, you need to follow a step-by-step process. Let's break it down:
Step 1: Expand the binomial (5a-b)^3 using the binomial theorem.
Using the binomial theorem, the expansion of (5a-b)^3 can be written as follows:
(5a-b)^3 = (5a)^3 - 3(5a)^2(b) + 3(5a)(b)^2 - (b)^3.
Simplifying further, we have:
(5a-b)^3 = 125a^3 - 75a^2b + 15ab^2 - b^3.
Step 2: Substitute the expanded expression into E=(2x+y)(125a^3 - 75a^2b + 15ab^2 - b^3).
By substituting (125a^3 - 75a^2b + 15ab^2 - b^3) into the expression E=(2x+y), we get:
E = (2x+y)(125a^3 - 75a^2b + 15ab^2 - b^3).
Step 3: Distribute the terms within the parentheses.
Using the distributive property, we multiply each term in (2x+y) by every term in (125a^3 - 75a^2b + 15ab^2 - b^3). We get:
E = (2x)(125a^3 - 75a^2b + 15ab^2 - b^3) + (y)(125a^3 - 75a^2b + 15ab^2 - b^3).
Step 4: Simplify the expression.
Now, we can simplify each term in the expanded expression:
First term: (2x)(125a^3 - 75a^2b + 15ab^2 - b^3) = 250xa^3 - 150xa^2b + 30xab^2 - 2xb^3.
Second term: (y)(125a^3 - 75a^2b + 15ab^2 - b^3) = 125ya^3 - 75ya^2b + 15yab^2 - yb^3.
Step 5: Combine like terms to get the final expression.
Now, combine like terms from the previous step:
E = 250xa^3 - 150xa^2b + 30xab^2 - 2xb^3 + 125ya^3 - 75ya^2b + 15yab^2 - yb^3.
Therefore, the final algebraic expression is E = 250xa^3 - 150xa^2b + 30xab^2 - 2xb^3 + 125ya^3 - 75ya^2b + 15yab^2 - yb^3.