Problem 4 - Algebra 2 and Trigonometry

Question

Perform the indicated operations and write the result in simplest form. 6\(c^{2} d\left(-2 c d^{3}\right)\)

Step-by-Step Solution

Answer

The result is \(-12c^3d^4\).

Step 1: Understand the Operation Indicated

We need to perform the multiplication of two expressions: \(6c^2d\) and \(-2cd^3\). This means multiplying both the coefficients and like variables separately.

Step 2: Multiply the Coefficients

The coefficients are the numerical parts of the expressions, which are 6 and -2. Multiply them together: \(6 \times (-2) = -12\).

Step 3: Apply the Power Rule for Multiplication

According to the power rule, when multiplying variables with the same base, you add their exponents. Multiply the \(c\) terms: \(c^2 \cdot c = c^{2+1} = c^3\). Multiply the \(d\) terms: \(d \cdot d^3 = d^{1+3} = d^4\).

Step 4: Write the Result in Simplest Form

Combine the results from the previous steps. The simplified form of the expression is: \(-12c^3d^4\).

Key Concepts

Coefficients

In polynomial multiplication, coefficients play a pivotal role. They are the numerical parts in terms and guide us on how much of each variable's term is included.

• When multiplying polynomials, you multiply coefficients first.
• In the example provided, the coefficients were 6 and -2.
• These are multiplied together to get -12.

The product of coefficients gives the new coefficient for the result. Coefficients are crucial because they scale the effect of the variables in an expression. By understanding how to multiply them, you handle a significant part of polynomial multiplication.

Power Rule

Applying the power rule is essential when multiplying polynomials with similar variables. This rule tells us how to handle the exponents:

• When you multiply like bases, such as the variable \(c\), you add their exponents.
• In our example, \(c^2\) was multiplied by \(c\), resulting in \(c^{2+1} = c^3\).

Variable Multiplication

The power rule simplifies the task of managing exponents during multiplication:

• Remember to apply it separately to each variable with the same base.
• Doing so simplifies variable expressions considerably in polynomial multiplication.

Exponents

Exponents indicate how many times a base is multiplied by itself, and they serve a special function in polynomial expressions:

• They simplify expressions and make calculations algorithmically manageable.
• In our exercise, although there is no initial exponent on \(d\), it is implicit as \(d^1\).
• Thus, \(d^1\) multiplied by \(d^3\) gives \(d^4\).

Exponent Handling

Understanding exponents helps manage multiplications in polynomials efficiently:

• Add the exponents when the same bases are multiplied.
• Always identify implicit exponents, such as \(x\) being \(x^1\).