Problem 24 - Algebra and Trigonometry

Question

Perform the indicated operations and simplify. $$ 3 x^{3}\left(x^{4}-4 x^{2}+5\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(3x^7 - 12x^5 + 15x^3\).

Step 1: Understand the Expression

The given expression is \( 3x^3(x^4 - 4x^2 + 5) \). We need to distribute \(3x^3\) across each term inside the parentheses to simplify the expression.

Step 2: Distribute \(3x^3\) to the First Term

Distribute \(3x^3\) to the first term \(x^4\). This results in: \[ 3x^3 imes x^4 = 3x^{3+4} = 3x^7 \].

Step 3: Distribute \(3x^3\) to the Second Term

Distribute \(3x^3\) to the second term \(-4x^2\). This calculation gives us:\[ 3x^3 imes (-4x^2) = -12x^{3+2} = -12x^5 \].

Step 4: Distribute \(3x^3\) to the Third Term

Distribute \(3x^3\) to the third term \(+5\). The resulting expression is:\[ 3x^3 imes 5 = 15x^3 \].

Step 5: Combine All Components

After distributing the terms, combine all components from the previous steps to form the simplified expression:\[ 3x^7 - 12x^5 + 15x^3 \]. This is the expression simplified.

Key Concepts

Distributive Property

When dealing with polynomial expressions, the distributive property is a key concept that helps you simplify expressions. This property states that you can distribute or multiply a single term across each term in a parenthesis. For instance, consider an expression like \( a(b+c) \). Applying the distributive property, you multiply \( a \) with each term inside the parenthesis, resulting in \( ab + ac \).

In the exercise at hand, we have \( 3x^3(x^4 - 4x^2 + 5) \). According to the distributive property, we need to multiply \( 3x^3 \) with each of the terms \( x^4, -4x^2, \) and \( 5 \).

• Multiply \( 3x^3 \) with \( x^4 \) to get \( 3x^7 \).
• Multiply \( 3x^3 \) with \( -4x^2 \) to get \( -12x^5 \).
• Multiply \( 3x^3 \) with \( 5 \) to get \( 15x^3 \).

This approach allows you to break down and simplify complex polynomial expressions efficiently.

Polynomial Multiplication

Polynomial multiplication involves multiplying each term in one polynomial by each term in another. It requires attention to detail to ensure each term is correctly multiplied.

In our exercise, we're simply multiplying a monomial \( 3x^3 \) with a trinomial \( x^4 - 4x^2 + 5 \). To do this effectively:

• Distribute the monomial to each term in the trinomial.
• Multiply the coefficients (numbers) and add the exponents of the like bases.

For example, when you multiply \( 3x^3 \) with \( x^4 \), you multiply 3 by the implied coefficient 1 (giving 3), and since the bases are the same (x), you add the exponents: \( 3x^3 \times x^4 = 3x^{3+4} = 3x^7 \).

This careful multiplication process ensures that the polynomial expression stays accurate as you simplify it.

Exponent Rules

Exponent rules are essential for simplifying expressions that involve powers of variables. These rules help us to deal with operations like adding, subtracting, or multiplying exponents, which occur frequently during polynomial multiplication.

The most important exponent rule used in the exercise is when multiplying like bases, you add the exponents. Consider two terms, \( x^a \) and \( x^b \): when you multiply them, you obtain \( x^{a+b} \).

During the simplification of \( 3x^3(x^4 - 4x^2 + 5) \), you apply this rule:

• For \( x^3 \times x^4 \), add the exponents: \( 3+4 \) to get \( x^7 \).
• For \( x^3 \times x^2 \), add 3 and 2, giving you \( x^5 \).

By applying these rules, you ensure each step of your calculation aligns with mathematical principles. This leads to accurate simplification and understanding of polynomial expressions.