Find each product. See Examples 3–5. x^2(3x-2)(5x+1)

Verified step by step guidance

1. Distribute \(x^2\) to each term in the first binomial \((3x - 2)\), resulting in \(x^2 \cdot 3x\) and \(x^2 \cdot (-2)\).
2. Simplify the expressions from the distribution: \(x^2 \cdot 3x = 3x^3\) and \(x^2 \cdot (-2) = -2x^2\).
3. Now, multiply the result \((3x^3 - 2x^2)\) by the second binomial \((5x + 1)\).
4. Distribute each term of \((3x^3 - 2x^2)\) to each term in \((5x + 1)\), starting with \(3x^3 \cdot 5x\), \(3x^3 \cdot 1\), \(-2x^2 \cdot 5x\), and \(-2x^2 \cdot 1\).
5. Combine like terms from the expanded expression to simplify the polynomial.

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves multiplying two or more polynomials together to form a new polynomial. This process requires distributing each term in one polynomial to every term in the other polynomial, ensuring that all combinations are accounted for. For example, when multiplying (3x - 2) by (5x + 1), each term in the first polynomial is multiplied by each term in the second.

Distributive Property

The distributive property is a fundamental algebraic principle that states a(b + c) = ab + ac. This property is essential when expanding polynomials, as it allows for the systematic distribution of terms. In the context of the given expression, it helps in breaking down the multiplication of the polynomial factors into manageable parts.

Combining Like Terms

Combining like terms is the process of simplifying an expression by adding or subtracting terms that have the same variable raised to the same power. This step is crucial after performing polynomial multiplication, as it helps to consolidate the expression into its simplest form. For instance, after expanding the product, any terms with the same degree can be combined to streamline the final polynomial.