Find each product : (3x-2)(4x^2+3x-5)

Verified step by step guidance

1. Distribute the first term of the first polynomial, $3x$, to each term in the second polynomial $(4x^2+3x-5)$.
2. Multiply $3x$ by $4x^2$ to get $12x^3$.
3. Multiply $3x$ by $3x$ to get $9x^2$.
4. Multiply $3x$ by $-5$ to get $-15x$.
5. Repeat the distribution process with the second term of the first polynomial, $-2$, and add the results to the previous products.

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves distributing each term in one polynomial to every term in another polynomial. This process, often referred to as the distributive property, ensures that all combinations of terms are accounted for, leading to a new polynomial that combines like terms. For example, in multiplying (3x - 2) by (4x^2 + 3x - 5), each term in the first polynomial must be multiplied by each term in the second.

Combining Like Terms

Combining like terms is a fundamental algebraic process where terms that have the same variable raised to the same power are added or subtracted. This simplification is crucial after performing operations like multiplication, as it helps to express the polynomial in its simplest form. For instance, after multiplying the polynomials, you may end up with several terms that can be combined to streamline the expression.

Standard Form of a Polynomial

The standard form of a polynomial is a way of writing the polynomial such that the terms are arranged in descending order of their degree. This format makes it easier to analyze the polynomial's properties, such as its degree and leading coefficient. For example, after finding the product of (3x - 2)(4x^2 + 3x - 5), the result should be organized from the highest degree term to the constant term.