Find each product. $2b^3(b^2-4b+3)$

Verified step by step guidance

1. Identify the expression to be multiplied: \$2b^3(b^2 - 4b + 3)$.
2. Apply the distributive property by multiplying \$2b^3$ with each term inside the parentheses separately.
3. Multiply \$2b^3\( by the first term: \)b^2$. This gives \(2b^3 \times b^2 = 2b^{3+2} = 2b^5\).
4. Multiply \$2b^3\( by the second term: \)-4b$. This gives \(2b^3 \times (-4b) = -8b^{3+1} = -8b^4\).
5. Multiply \$2b^3\( by the third term: \(3\). This gives \(2b^3 \times 3 = 6b^3\). Then, write the product as the sum of these terms: \)2b^5 - 8b^4 + 6b^3$.

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves distributing each term in one polynomial to every term in the other polynomial. This process requires applying the distributive property to combine like terms and simplify the expression.

Exponent Rules

When multiplying terms with the same base, add their exponents. For example, b^3 multiplied by b^2 equals b^(3+2) = b^5. Understanding this rule is essential for correctly simplifying powers during multiplication.

Distributive Property

The distributive property states that a(b + c) = ab + ac. It allows you to multiply a single term by each term inside a parenthesis, which is crucial for expanding expressions like 2b^3(b^2 - 4b + 3).