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Chapter 5: Problem 7

Multiply. \(\left(-3.1 x^{3}\right)\left(4 x^{9}\right)\)

Short Answer

Expert verified
\(-12.4x^{12}\)

Step by step solution

01

Apply the Power of a Product Property

When multiplying terms with the same base, we need to multiply the coefficients and add the exponents of like bases. Let's use this rule on the expression.The expression given is \((-3.1) \cdot 4 \cdot x^{3} \cdot x^{9}\) Separate the numbers from the variables and multiply them separately.
02

Multiply the Coefficients

Take the numerical coefficients (\(-3.1\) and \(4\)) from the expression.Multiply them together:\(-3.1 \times 4 = -12.4\)So, the coefficient of the product is \(-12.4\).
03

Add the Exponents of x

With the property of exponents, when multiplying two powers that have the same base, you add the exponents.Here, we have the powers \(x^{3}\) and \(x^{9}\).Add the exponents: \(3 + 9 = 12\)The combined power becomes \(x^{12}\).
04

Combine the Results

Now, combine the coefficients and variable parts to form the final answer.The result of multiplying the coefficients is \(-12.4\) and the result of combining the exponents is \(x^{12}\).Thus, the expression becomes:\(-12.4 x^{12}\).

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Key Concepts

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

Coefficient Multiplication
When multiplying polynomials, a crucial step is coefficient multiplication. This involves the numerical values in front of the variables. For our example, the coefficients are \(-3.1\) and \(4\). Multiply these to get the coefficient of the resulting term:
  • Think of the coefficients like regular numbers. Multiply them the same way you would do with any numbers: \(-3.1 \times 4 = -12.4\).
  • Keep the sign in mind; a negative \((-3.1)\) times a positive \(4\) equals a negative result \(-12.4\).
This step ensures that you properly scale the magnitude of your polynomial product, keeping it accurate to the factors involved.
Exponent Addition
The operation of exponent addition is fundamental when multiplying terms with the same base. Exponents denote how many times a number is used as a factor. So, when you multiply \(x^3\) and \(x^9\), you simply add the exponents, because:
  • Both terms share the same base \(x\).
  • Add the exponents together: \(3 + 9 = 12\).
The result is an exponent of 12, thus forming the term \(x^{12}\). This gives the new power which shows how many times the base is multiplied by itself.
Power of a Product Property
The power of a product property simplifies expressions involving powers. When you have multiple terms with exponents and the same base, this property guides the multiplication process:
  • The property states that you multiply the coefficients and add the exponents of like bases.
  • In our example, for \(x^3 \cdot x^9\), by adding the exponents \(3 + 9\), we use this property to simplify to \(x^{12}\).
This property provides a neat, orderly way to handle multiplication with exponents, avoiding convolution and ensuring accuracy in algebraic expressions.

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