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Chapter 4: Problem 7

Find each product. $$ 14 x^{2} y^{3}\left(-2 x^{5} y\right) $$

Short Answer

Expert verified
-28x^{7}y^{4}

Step by step solution

01

Identify the Terms to Multiply

Recognize the terms in the given expression that need to be multiplied: \(14 x^{2} y^{3}\) and \(-2 x^{5} y\).
02

Multiply the Coefficients

Multiply the numerical coefficients: \(14\) and \(-2\). \[ 14 \times (-2) = -28 \]
03

Multiply the Variables with Exponents

Use the properties of exponents to multiply the variables. For the \(x\) terms, add the exponents: \(x^{2} \times x^{5} = x^{2+5} = x^{7}\). For the \(y\) terms, add the exponents: \(y^{3} \times y = y^{3+1} = y^{4}\).
04

Combine the Results

Combine the results from multiplying the coefficients and adding the exponents: \( -28x^{7}y^{4} \).

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Key Concepts

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

Exponents
Exponents indicate how many times a number or variable is multiplied by itself. For example, in the expression \( x^2 \), the variable \( x \) is multiplied by itself, resulting in \( x \times x \).
When multiplying expressions with the same base, you add their exponents.
In our exercise, we have: \( x^2 \times x^5 \). To multiply these, add the exponents: \[ x^{2+5} = x^7 \].
Similarly, for the \( y \) terms: \( y^3 \times y \) can be written as \[ y^{3+1} = y^4 \].
Remember, when there is no exponent shown next to a variable (as in \( y \)), it is implicitly \( y^1 \). This understanding helps simplify and correctly solve polynomial multiplications.
Product of Terms
When multiplying polynomials, we look for terms that can be combined using multiplication. This involves both the coefficients (numerical parts) and the variables (letters).
In the exercise, we need to multiply two terms: \( 14 x^2 y^3 \) and \( -2 x^5 y \).
  • First, multiply the coefficients: \( 14 \times -2 = -28 \).
  • Next, handle the variables. For the \( x \) terms, add the exponents: \( x^2 \times x^5 = x^{7} \).
  • Then, do the same for the \( y \) terms: \( y^3 \times y = y^4 \).
Combining these results, you get \( -28 x^7 y^4 \).
It's useful to use the distributive property to break down the steps into manageable parts, as this method helps ensure accuracy and improves understanding.
Coefficients
Coefficients are the numerical parts of terms in polynomial expressions, which stand in front of the variable part. They are multiplied together just like normal numbers.
In our specific example, the coefficients are \( 14 \) and \( -2 \).
Multiplying them together:
  • First multiply: \( 14 \times -2 \).
  • Since multiplying a positive number with a negative number results in a negative product, this gives us \( -28 \).
So, the coefficient for the final result is \( -28 \).
Understanding how to correctly multiply coefficients is crucial in managing and simplifying polynomial expressions efficiently.

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