Find the Product 3x(5x^2-x+4)

The question is asking for the result of multiplying a monomial, which is $3x$, by a trinomial, which is $(5x^2 - x + 4)$. The process involves using the distributive property (also known as the distributive law of multiplication) to multiply $3x$ by each term of the trinomial separately and then combining like terms if necessary. This will result in a polynomial that represents the product of the given expressions.

$3 x \left(\right. 5 x^{2} - x + 4 \left.\right)$

Answer

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Solution:

Step 1: Utilize the Distributive Property

Multiply $3x$ by each term inside the parentheses: $3x(5x^2) + 3x(-x) + 3x(4)$

Step 2: Commence Simplification

Step 2.1: Apply Commutative Property for Multiplication

Rearrange the terms: $3 \cdot 5 \cdot x \cdot x^2 + 3x(-x) + 3 \cdot x \cdot 4$

Step 2.2: Continue with Commutative Property

Further rearrange the terms: $3 \cdot 5 \cdot x \cdot x^2 + 3 \cdot (-1) \cdot x \cdot x + 3 \cdot x \cdot 4$

Step 2.3: Perform Multiplication of Constants

Multiply the constant $4$ with $3$: $3 \cdot 5 \cdot x \cdot x^2 + 3 \cdot (-1) \cdot x \cdot x + 12 \cdot x$

Step 3: Simplify Each Term

Step 3.1: Combine Exponents

Step 3.1.1: Position the $x^2$ term

Place $x^2$ next to $x$: $3 \cdot 5 \cdot (x^2 \cdot x) + 3 \cdot (-1) \cdot x \cdot x + 12 \cdot x$

Step 3.1.2: Multiply Powers of $x$

Step 3.1.2.1: Express $x$ as $x^1$

Write $x$ with an exponent: $3 \cdot 5 \cdot (x^2 \cdot x^1) + 3 \cdot (-1) \cdot x \cdot x + 12 \cdot x$

Step 3.1.2.2: Apply the Power Rule

Combine the exponents: $3 \cdot 5 \cdot x^{2+1} + 3 \cdot (-1) \cdot x \cdot x + 12 \cdot x$

Step 3.1.3: Sum the Exponents

Add the exponents $2$ and $1$: $3 \cdot 5 \cdot x^3 + 3 \cdot (-1) \cdot x \cdot x + 12 \cdot x$

Step 3.2: Multiply Constants

Multiply $3$ by $5$: $15 \cdot x^3 + 3 \cdot (-1) \cdot x \cdot x + 12 \cdot x$

Step 3.3: Combine Exponents of $x$

Step 3.3.1: Position the $x$ term

Place $x$ next to itself: $15 \cdot x^3 + 3 \cdot (-1) \cdot (x \cdot x) + 12 \cdot x$

Step 3.3.2: Multiply Powers of $x$

Multiply $x$ by itself: $15 \cdot x^3 + 3 \cdot (-1) \cdot x^2 + 12 \cdot x$

Step 3.4: Multiply Constants

Multiply $3$ by $-1$: $15 \cdot x^3 - 3 \cdot x^2 + 12 \cdot x$

Knowledge Notes:

The problem-solving process involves the following key mathematical concepts:

Distributive Property: This property states that $a(b + c) = ab + ac$. It allows us to multiply a single term by each term inside a set of parentheses.
Commutative Property of Multiplication: This property states that $ab = ba$. It allows us to rearrange the factors in a multiplication without changing the product.
Multiplication of Constants and Variables: When multiplying constants (numbers) or variables (like $x$), we simply multiply the numerical coefficients and add the exponents if the bases are the same.
Power Rule for Exponents: The power rule states that $a^m \cdot a^n = a^{m+n}$. When multiplying like bases, we add the exponents.
Simplification: This is the process of combining like terms and reducing expressions to their simplest form.

In this problem, we applied these concepts to multiply a monomial, $3x$, by a trinomial, $5x^2 - x + 4$, to find the product.