Problem 28 - Algebra for College Students

Question

Find each product. $$\left(-y^{3}\right)(-6 y)\left(-8 y^{4}\right)$$

Step-by-Step Solution

Answer

The product is \(-48y^8\).

Step 1: Identify the Coefficients

The coefficients in the expression are \(-1\), \(-6\), and \(-8\). To find the product of these coefficients, multiply them: \((-1) \times (-6) \times (-8)\).

Step 2: Multiply the Coefficients

Start by multiplying the first two coefficients: \((-1) \times (-6) = 6\). Then multiply the result by \(-8\): \(6 \times (-8) = -48\).

Step 3: Identify the Variables and Their Exponents

Each part of the expression has variables with exponents: \(y^3\), \(y^1\), and \(y^4\).

Step 4: Add the Exponents of the Same Variable

Combine the exponents of the variable \(y\) by adding them: \(3 + 1 + 4 = 8\). So, the product will have \(y^8\).

Step 5: Combine the Coefficients and Variables

Combine the calculated coefficient with the variable expression to obtain the final result: \(-48y^8\).

Key Concepts

Coefficients

Coefficients are the numerical parts of a term in a polynomial. In the expression \((-y^3)(-6y)(-8y^4)\), the coefficients are the numbers in front of the variables:

• each term:
  • -1 (from \(-y^3\)),
  • -6 (from \(-6y\)),
  • and -8 (from \(-8y^4\)).

Each coefficient gives you a way to measure how the variable part of the term contributes to the expression. When multiplying terms, you simply multiply these numbers together.

• First, multiply -1 and -6 to get 6.
• Then, multiply 6 by -8 to get the final product: -48.

This process of multiplying the coefficients helps simplify the numerical part of the final expression.

Exponents

Exponents are a shorthand way to express repeated multiplication of the same number. In polynomials, they describe how many times a variable is multiplied by itself. For instance, in the expression \(y^4\), the 4 is an exponent, meaning you multiply \(y\) by itself 4 times: \(y \times y \times y \times y\). To find the product of polynomials, like \((-y^3)(-6y)(-8y^4)\), you add the exponents of like variables:

• The exponents are 3, 1, and 4 for the variable \(y\).
• To combine them, add: \(3 + 1 + 4 = 8\).
• This results in \(y^8\).

By understanding exponents, you can efficiently multiply variables without writing out lengthy multiplications.

Product of Polynomials

The product of polynomials involves finding the result of multiplying two or more polynomial expressions. This process is a fundamental part of working with algebraic expressions and helps reveal new expressions that are equivalent to the originals in a simplified form.In the given problem, we are multiplying three terms:

• \((-y^3)(-6y)(-8y^4)\).

Here's how you can break it down:

• First, focus on the coefficients: Multiply \((-1)\), \((-6)\), and \((-8)\).
• Then, handle the exponents for the variable \(y\): Combine the powers \(3 + 1 + 4\).
• Finally, join the results: The final expression is \(-48y^8\).

Using this method allows you to manage complex expressions in a straightforward way, leading to a single simplified polynomial.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations such as addition, subtraction, multiplication, or division. These expressions form the building blocks of algebra. In working with these expressions, such as \((-y^3)(-6y)(-8y^4)\), it's important to recognize:

• The structure of terms: Each includes a coefficient \(-1, -6, -8\), a variable \(y\), and exponents \(3, 1, 4\).
• The operations being used: In this case, primarily multiplication.
• The simplified expression: \(-48y^8\), representing the unified expression.

By understanding and manipulating algebraic expressions, you'll be able to solve equations, handle complex problems, and apply these skills in various math and science disciplines.