Find each product. $x^2(3x-2)(5x+1)$

Verified step by step guidance

1. Identify the expression to be multiplied: \(x^2(3x - 2)(5x + 1)\).
2. First, multiply the two binomials \((3x - 2)\) and \((5x + 1)\) using the distributive property (FOIL method): multiply each term in the first binomial by each term in the second binomial.
3. Write out the multiplication: \((3x)(5x) + (3x)(1) + (-2)(5x) + (-2)(1)\).
4. Simplify each term: \$15x^2 + 3x - 10x - 2\(, then combine like terms to get \)15x^2 - 7x - 2$.
5. Finally, multiply the resulting trinomial by \(x^2\): \(x^2(15x^2 - 7x - 2)\) by distributing \(x^2\) to each term inside the parentheses.

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves multiplying two or more polynomials by distributing each term in one polynomial to every term in the other. This process requires applying the distributive property and combining like terms to simplify the expression.

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 essential when expanding expressions like x^2(3x - 2)(5x + 1).

Combining Like Terms

After multiplying polynomials, you often get terms with the same variable raised to the same power. Combining like terms means adding or subtracting these terms to simplify the expression into its standard polynomial form.