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Chapter 5: Problem 22

Multiply. $$ 7 x^{2}\left(-2 x^{3}\right)\left(2 x^{6}\right) $$

Short Answer

Expert verified
-28 x^{11}

Step by step solution

01

Understand the expression

The given expression is \( 7 x^{2} \times (-2 x^{3}) \times (2 x^{6}) \). Notice that there are coefficients and variables with exponents.
02

Multiply the coefficients

First, focus on the numerical coefficients: \(7 \times (-2) \times 2 \). Calculate this product: \(7 \times (-2) = -14\) and then \(-14 \times 2 = -28\). So, the coefficient of the final product will be \(-28\).
03

Multiply the variables

Next, handle the variables. When multiplying variables with exponents, add the exponents: \( x^{2} \times x^{3} \times x^{6} = x^{(2+3+6)} = x^{11} \).
04

Combine the results

Combine the result of the coefficients and the variables: The product is \( -28 x^{11} \).

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Key Concepts

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

Coefficients
When multiplying algebraic expressions, it's crucial to handle the coefficients correctly.
Coefficients are the numerical parts of the terms in an algebraic expression. For example, in the term $$ 7x^2, $$ 7 is the coefficient.
To multiply coefficients:
  • Combine the numerical values through multiplication.
    For instance, in our exercise, the coefficients given are 7, -2, and 2.
    We first multiply 7 by -2 to get -14 and then -14 by 2 to get -28. Thus, the final coefficient result is -28.
Understanding how to correctly multiply coefficients is a foundational step in solving algebraic expressions.
Exponents
Exponents indicate how many times a number (the base) is multiplied by itself.
In expressions like $$x^2$$, the exponent is 2, indicating that x is multiplied by itself two times.
When multiplying variables with exponents, we use the rule of adding exponents.
For example, in the exercise $$ 7x^2 \times (-2x^3) \times (2x^6), $$ multiply the bases while adding the exponents:
  • Here, the exponents are 2, 3, and 6. We add them: $$ 2 + 3 + 6 = 11. $$
The variable term in our final product will be $$ x^{11}. This step is essential to ensure that the resulting expressions are correctly simplified.
Polynomials
Polynomials are algebraic expressions that include coefficients and variables.
They can have one or more terms. An example of a polynomial could be $$ 3x^2 + 2x + 1.$$ Understanding polynomials involves:
  • Recognizing the terms: Each term in a polynomial includes a coefficient and a variable with an exponent.
  • Combining like terms: This means adding or multiplying the terms that have the same variable raised to the same power. However, in our exercise, no like terms exist among the individual multiplications.
The exercise we solved involves multiplying a polynomial with multiple terms. This process combines the coefficients and adds the exponents of like variables, thereby simplifying the expression. The final product in this case is $$ -28x^{11}, $$ a single term polynomial with a coefficient of -28 and an exponent of 11.

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