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Chapter 5: Problem 261

Find each product. $$ \left(\frac{1}{4} d^{5}\right)\left(36 d^{2}\right) $$

Short Answer

Expert verified
9d^{7}

Step by step solution

01

- Multiply the coefficients

First, multiply the numerical coefficients of the two terms. The coefficients are \(\frac{1}{4}\) and 36. \(\frac{1}{4} \times 36 = 9\).
02

- Combine the variables

Next, multiply the variables by adding their exponents. The variables are \(d^{5}\) and \(d^{2}\). According to the laws of exponents, \(d^{5} \times d^{2} = d^{5+2} = d^{7}\).
03

- Combine the results

Finally, combine the results from the previous steps. The product is \(9d^{7}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coefficients
Coefficients are the numerical part of a term in a polynomial. They are the numbers situated before the variable. In our example, \(\frac{1}{4} d^{5}\) and \(36 d^{2}\), the coefficients are \(\frac{1}{4}\) and 36 respectively.
To find the product of these terms, the first step is to multiply the coefficients.
This is done by multiplying \(\frac{1}{4}\) with 36. \(\frac{1}{4} \times 36 = 9\). Understanding coefficients is crucial because they determine the scaling of the term in the polynomial.
So, once you perform this multiplication, you get the coefficient of the resulting polynomial term.
Laws of Exponents
The laws of exponents help us deal with powers of variables efficiently. When multiplying variables with the same base, we add their exponents. This is one of the key laws.
In our example, we have \(d^5\) and \(d^2\).
The base here is the variable \(d\). According to the law, when we multiply these, we add the exponents: \(d^{5} \times d^{2} = d^{5+2} = d^{7}\).
These laws simplify expressions and are fundamental for solving polynomial multiplication.
Combining Like Terms
Combining like terms is the process of adding or subtracting terms that have the same variable raised to the same power. This concept is important when simplifying expressions.
In the provided example, after multiplying the coefficients and the variables separately, we combined them to get the final result.
Here, there were no like terms to combine, but typically, combining like terms reduces and simplifies the polynomial.
Understanding how to identify and combine like terms boosts efficiency in algebraic manipulations and managing polynomials.

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Most popular questions from this chapter

Multiply each pair of conjugates using the Product of Conjugates Pattern. $$ (5 k+6)(5 k-6) $$

Multiply or divide as indicated. Write your answer in decimal form. (a) \(\left(3.5 \times 10^{-4}\right)\left(1.6 \times 10^{-2}\right)\) (?) \(\frac{8 \times 10^{6}}{4 \times 10^{-1}}\)

Multiply. Use either method. $$ (w-7)\left(w^{2}-9 w+10\right) $$

Find each product. $$ (2 x+y)(x-2 y) $$

For functions \(\quad f(x)=x+2\) and \(g(x)=3 x^{2}-2 x+4,\) find (a) \((f \cdot g)(x)\) (b) \((f \cdot g)(-1)\)
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