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Celine's Cereal Company is launching a new brand of cereal, and she is considering two different sizes for the base of the boxes.



Box 1: The length is 3 times the width.

Box 2: The length is 1 less than 4 times the width.



$\text{length} \cdot \text{width} \cdot \text{height} = (4x - 1) \cdot x \cdot x^3$

$\text{volume} = 4x^5 - x^4$



Degree of length: [ ]

Degree of width: [ ]

Degree of height: [ ]

Degree of volume: [ ]



The [ ] of the degrees of each factor is the degree of the product.



Why does this make sense?

[ ]

Verified Answer :

  • The length \((4x-1)\) has degree 1, the width \((x)\) has degree 1, and the height \((x^3)\) has degree 3.
  • The product volume \((4x^5-x^4)\) has degree 5.
  • The sum of the degrees of the factors \((1+1+3)\) equals the degree of the product (5).
  • The highest power of a product is the sum of the highest powers of its factors due to the rule of multiplying powers of the same base.
  • Final degrees: Degree of length: \(\boxed{1}\), Degree of width: \(\boxed{1}\), Degree of height: \(\boxed{3}\), Degree of volume: \(\boxed{5}\). The sum of the degrees of each factor is the degree of the product.

Explanation

  1. Data and Problem Analysis

We are given the expressions for the dimensions of a cereal box, where the width is \(x\), the length is \(4x-1\), and the height is \(x^3\). The volume \(V(x)\) is the product of these three dimensions:
\[V(x) = \text{length} \times \text{width} \times \text{height} = (4x - 1) \cdot x \cdot x^3\]
By multiplying these terms, we obtain the expression for the volume:
\[V(x) = (4x - 1) \cdot x^4 = 4x^5 - x^4\]
We need to determine the degree of each factor and the degree of the resulting product polynomial, then explain the relationship between these degrees.

  1. Determining the Degrees of the Polynomials

The degree of a polynomial is the highest power of the variable \(x\) present in the expression.

  • For the length, \(L(x) = 4x - 1\), the highest power of \(x\) is 1. Thus, the degree is 1.
  • For the width, \(W(x) = x\), the highest power of \(x\) is 1. Thus, the degree is 1.
  • For the height, \(H(x) = x^3\), the highest power of \(x\) is 3. Thus, the degree is 3.
  • For the volume, \(V(x) = 4x^5 - x^4\), the highest power of \(x\) is 5. Thus, the degree is 5.
  1. Explaining the Relationship Between Degrees

Comparing the degrees, we see that the sum of the degrees of the individual factors is:
\[1 (\text{length}) + 1 (\text{width}) + 3 (\text{height}) = 5\]
This sum matches the degree of the product polynomial (the volume).
This rule works because, when multiplying polynomials, the term with the highest power in the resulting product is created by multiplying the terms with the highest powers from each individual factor. According to the laws of exponents, when we multiply variables with exponents, we add the exponents together (e.g., \(x^a \cdot x^b \cdot x^c = x^{a+b+c}\)). Therefore, the highest exponent in the product is simply the sum of the highest exponents of the factors.

Examples

In engineering and manufacturing, understanding the degree of polynomials is essential for scaling models. If an engineer knows that the volume of a container depends on three dimensions, each expressed as a polynomial of a specific degree, they can immediately predict the complexity and growth rate of the volume function. For instance, if they double the dimensions (scaling \(x\) by a factor of 2), a volume function of degree 5 will grow by a factor of \(2^5 = 32\), which is critical for planning material costs and storage capacity.

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