Question

Multiply \(2 x^4 \cdot 4 x^5\). A. \(6 x^9\) B. \(8 x^9\) C. \(6 x^{20}\) D. \(8 x^{20}\)

Short answer

B. \(8 x^9\)

Step-by-step solution

Multiply the coefficients

Identify the coefficients of each term that need to be multiplied. For the terms \(2 x^4\) and \(4 x^5\), the coefficients are 2 and 4 respectively. Multiply these coefficients together: \(2 \times 4 = 8\).

Add the exponents of like bases

Next, observe the bases of both terms. The base in both terms is \(x\). When multiplying terms with the same base, the exponents are added together. For \(x^4\) and \(x^5\), add the exponents: \(4 + 5 = 9\).

Combine the results

Now combine the multiplied coefficients and the new exponent: \(8 x^9\).

Key concepts

coefficients

To understand polynomial multiplication, it's vital to first grasp the concept of coefficients. A coefficient is a numerical value placed in front of a variable in a polynomial. In the exercise, we had to multiply the coefficients from two terms: 2 from the term \(2x^4\) and 4 from the term \(4x^5\). To find the resulting coefficient, we multiplied 2 and 4, which equals 8.
Here’s a quick recap:
- Identify the numerical values in front of each variable.
- Multiply these values together to find the new coefficient.
By mastering coefficients, polynomial multiplication can become much easier.

exponents

When working with polynomials, exponents indicate how many times a base (like x) is multiplied by itself. In the exercise, you saw terms like \(x^4\) and \(x^5\). The small numbers, 4 and 5, are the exponents. To multiply these terms, remember that the base in both terms is the same (x). In such scenarios, you add the exponents together:
- For \(x^4\) (which means x multiplied by itself 4 times) and \(x^5\) (which means x multiplied by itself 5 times), you simply add the exponents: \(4 + 5 = 9\).
- The result, thus, is \(x^9\).
Summarizing it, when multiplying like bases, always add the exponents to get the new exponent.

like bases

Like bases refer to variables that are the same (e.g., x, y). In our example, both terms had the base x: \(2x^4\) and \(4x^5\). To multiply terms with like bases:
- First, ensure the variables are the same.
- Multiply their coefficients together.
- Add their exponents together.
This rule simplifies polynomial multiplication significantly. Remember: only add exponents if the bases are identical. This principle keeps operations neat and correct. Knowing how to handle like bases makes many algebraic problems much easier to solve. Always look for the same variable when adding exponents.