Problem 2 - Algebra for College Students

Question

Find each product. $$\left(6 x^{3}\right)\left(7 x^{2}\right)$$

Step-by-Step Solution

Answer

The product is \(42x^5\).

Step 1: Identify the Coefficients and Exponents

The given expression is \((6x^3) \times (7x^2)\). Here, the coefficients are 6 and 7, while the exponents of the variable \(x\) are 3 and 2.

Step 2: Multiply the Coefficients

Multiply the coefficients 6 and 7. Thus, \(6 \times 7 = 42\).

Step 3: Add the Exponents

When multiplying powers with the same base, you add their exponents. So, for \(x^3\) and \(x^2\), add the exponents: \(3 + 2 = 5\).

Step 4: Write the Final Product

Combine the product of the coefficients and the sum of the exponents to form the final expression: \(42x^5\).

Key Concepts

Coefficients

Coefficients are the numerical factors in terms of an algebraic expression. When dealing with polynomial multiplication, identifying the coefficients is an essential first step. In the expression \((6x^3) \times (7x^2)\), the coefficients are 6 and 7. Coefficients tell us "how many" of each base unit we have.

• The term \(6x^3\) means 6 times whatever \(x^3\) is.
• Similarly, \(7x^2\) means 7 times \(x^2\).

When multiplying polynomials, you multiply the coefficients first before dealing with the variables. For our example:

• Multiply 6 by 7 to get 42.

So, the product's coefficient is 42. This serves as the constant factor in the final simplified expression.

Exponents

Exponents represent the power to which a number or variable is raised. They play a crucial role in defining how expressions expand during multiplication. They also indicate repeated multiplication of a number by itself. In the expression \(6x^3\), the exponent on \(x\) is 3, while for \(7x^2\), it is 2.

• An exponent of 3 means the base \(x\) is multiplied by itself 3 times (\(x \times x \times x\)).
• For an exponent of 2, the base \(x\) is multiplied by itself twice (\(x \times x\)).

When multiplying terms with the same base, add the exponents together. This is a principle of exponentiation:

• For \(x^3 \times x^2\), add 3 and 2 to get 5.

Thus, \(x^5\) becomes the new term in the product, adding depth to the polynomial.

Distributive Property

The distributive property is a fundamental algebraic principle and an essential tool in polynomial multiplication. It involves multiplying each term within a parenthesis by every term in another parenthesis. This property simplifies multiplication processes by distributing the operations.
In the given exercise, we're not directly using the distributive property since each term within a parenthesis is straightforwardly multiplied by the other. However, understanding this property is crucial, especially with complex polynomials.
For instance, if our exercise included a more complicated expression like \((a + b) \times (c + d)\), the distributive property would require calculating:

• \(a \times c\)
• \(a \times d\)
• \(b \times c\)
• \(b \times d\)

Then, you would sum all these products. This principle ensures efficient organization, helping avoid mistakes while multiplying polynomials.