Adding and Multiplying Polynomials: Solving Algebraic Expressions

How do we solve algebraic expressions involving polynomials?

Given the expression: (-4b^2 + 8b) (-4b^3 + 5b^2 - 8b), how can we accurately find the answer?

Solution: Exploring Polynomial Operations

To solve algebraic expressions involving polynomials, we need to understand the fundamental operations of adding and multiplying polynomials. Let's break down the process step by step:

When dealing with polynomials, it's essential to differentiate between addition and multiplication. In this case, we are given the expression (-4b^2 + 8b) (-4b^3 + 5b^2 - 8b), which indicates a multiplication operation.

For multiplication of polynomials, we need to distribute each term in one polynomial across all terms in the other polynomial. Let's apply this concept to our expression:

Multiplication of Polynomials:

(-4b^2 + 8b) (-4b^3 + 5b^2 - 8b)

Distribute each term in the first polynomial across all terms in the second polynomial:

-4b^2 * -4b^3 + (-4b^2 * 5b^2) + (-4b^2 * -8b) + 8b * -4b^3 + (8b * 5b^2) + (8b * -8b)

Combine like terms and simplify the expression:

16b^5 - 20b^4 + 32b^3 - 32b^4 + 40b^3 - 64b^2

Final Answer: 16b^5 - 52b^4 + 72b^3 - 64b^2

Therefore, by correctly applying the principles of polynomial multiplication, we have successfully solved the given algebraic expression and obtained the expanded polynomial in standard form.