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Chapter 5: Problem 45

Find each product. $$-3 x^{2} \cdot 5 x^{4}$$

Short Answer

Expert verified
-15x^{6}

Step by step solution

01

- Identify coefficients and variables

First, identify the coefficients and the variables in each term. In the expression ewline newline$$-3x^{2} \times 5x^{4}$$ewline the coefficients are -3 and 5, and the variables are \(x^{2}\) and \(x^{4}\).
02

Multiply the coefficients

Multiply the coefficients (-3) and (5) together. $$-3 \times 5 = -15.$$
03

Apply laws of exponents

When multiplying like bases, add the exponents. For the variablesewline ewline$$x^{2} \times x^{4} = x^{2+4} = x^{6}.$$
04

Combine the results

Now combine the results from the previous steps. The product of the coefficients is -15, and the product of the variables is \(x^{6}\). Thus, your final answer is: $$-15x^{6}.$$

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Key Concepts

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

coefficients
In polynomial multiplication, a key concept is working with coefficients. Coefficients are the numerical factors in terms of a polynomial. For example, in the term \(-3x^{2}\), -3 is the coefficient. These numbers tell us how many times to multiply the variable(s).

When multiplying coefficients, simply perform regular multiplication. In our exercise, the coefficients are -3 and 5. We multiply them as follows:
  • -3 \(\times\) 5 = -15

This result gives us the coefficient of the resulting term. Remember, the signs of the numbers matter too. A negative times a positive gives a negative result.
exponents
Exponents represent the number of times a base is multiplied by itself. They appear as small numbers to the upper right of the base. For instance, in \(x^{2}\), 2 is the exponent, meaning x is multiplied by itself 2 times (x \(\times\) x).

When multiplying terms with the same base, you add the exponents together. In our problem, we have:
  • \(x^{2} \times x^{4} = x^{2+4} = x^{6}\)

Here, both terms have the same base x. Adding their exponents (2 and 4) gives us 6. Thus, \(x^{2} \times x^{4}\) becomes \(x^{6}\).
laws of exponents
The laws of exponents are rules that simplify expressions involving exponents. One of the crucial laws states that when you multiply two exponents with the same base, you add the exponents. The law can be written as:
  • \(a^{m} \times a^{n} = a^{m+n}\)

In our exercise, we applied this law:
  • \(x^{2} \times x^{4} = x^{2+4} = x^{6}\)

This law helps in reducing the complexity of polynomial multiplication. Understanding these rules allows you to work with more advanced algebraic expressions easily. It makes the process quicker and less error-prone.

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Most popular questions from this chapter

On an exam, a student factored \(2 x^{2}-6 x+4\) as \((2 x-4)(x-1) .\) Even though the student carefully checked that $$ (2 x-4)(x-1)=2 x^{2}-6 x+4 $$ the student lost some points. What went wrong?

Factor each polynomial using the trial-and-error method. $$ 3 x^{2}+25 x+8 $$

Factor each polynomial by grouping. $$ t^{2} a-3 a-3+t^{2} $$

Factor each polynomial and explain how you decided which method to use. a) \(x^{2}+10 x+25\) b) \(x^{2}-10 x+25\) c) \(x^{2}+26 x+25\) d) \(x^{2}-25\) e) \(x^{2}+25\)

Firing an \(M-16 .\) If an M- 16 is fired straight upward, then the height \(h(t)\) of the bullet in feet at time \(t\) in seconds is given by $$h(t)=-16 t^{2}+325 t$$ a) What is the height of the bullet 5 seconds after it is fired? b) How long does it take for the bullet to return to the earth?
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