Problem 66 - Algebra Form and Function

Question

Give polynomials satisfying the given conditions if possible, or say why it is impossible to do so. Two polynomials whose product has degree \(9 .\)

Step-by-Step Solution

Answer

Question: Provide examples of two polynomials that multiply together to give a product with degree 9. Answer: One example is the pair of polynomials P(x) = x^4 and Q(x) = x^5. Their product will be (P · Q)(x) = x^(4+5) = x^9. Note that there are many other pairs of polynomials with different degrees that also satisfy the given condition, such as P(x) = x^3 and Q(x) = x^6.

Step 1: Determine the degrees of the polynomials

To find two polynomials that multiply together to give a product with degree \(9\), we need to find two degrees that add up to \(9\). For instance, the degrees of the polynomials could be \(1\) and \(8\), or \(2\) and \(7\), or \(3\) and \(6\), or \(4\) and \(5\). Any of these pairs of degrees will satisfy the given condition.

Step 2: Example of suitable polynomials

We will provide examples of polynomials for each pair of degrees: 1. If the degrees of the polynomials are \(1\) and \(8\), the polynomials could be: \[P(x) = x\] \[Q(x) = x^8\] Their product will be: \[(P \cdot Q)(x) = x^{1+8} = x^9\] 2. If the degrees of the polynomials are \(2\) and \(7\), the polynomials could be: \[P(x) = x^2\] \[Q(x) = x^7\] Their product will be: \[(P \cdot Q)(x) = x^{2+7} = x^9\] 3. If the degrees of the polynomials are \(3\) and \(6\), the polynomials could be: \[P(x) = x^3\] \[Q(x) = x^6\] Their product will be: \[(P \cdot Q)(x) = x^{3+6} = x^9\] 4. If the degrees of the polynomials are \(4\) and \(5\), the polynomials could be: \[P(x) = x^4\] \[Q(x) = x^5\] Their product will be: \[(P \cdot Q)(x) = x^{4+5} = x^9\] In conclusion, there are many pairs of polynomials whose product has degree \(9\). We have provided a few examples above. Note that these are not unique solutions - any polynomials with the correct degrees will satisfy the given condition.

Key Concepts

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. It tells you how many times the variable multiplies itself in the polynomial. For example, in the polynomial \(x^3 + 2x^2 - x + 7\), the degree is 3 because the highest exponent of \(x\) is 3.

To better understand this, let's break it down:

• The degree is important because it gives insight into the polynomial's behavior, such as the number of roots it could have.
• A polynomial with a degree of \(n\) can have up to \(n\) roots or solutions.
• When adding or subtracting polynomials, the degree of the result usually equals the highest degree among the terms involved.

Knowing the degree of a polynomial is crucial for operations like multiplication or finding polynomial products.

Polynomial Product

The polynomial product refers to the result of multiplying two or more polynomials together. It's an essential concept in algebra as it helps in simplifying and solving equations. When polynomials are multiplied, the exponents of like terms are added, which determines the degree of the resulting polynomial.

To calculate a polynomial product, use the distributive property:

• Multiply each term in the first polynomial by each term in the second polynomial.
• Combine the results and simplify by adding coefficients of like terms.

For example, if you have two polynomials \(P(x) = x^2 + 2x + 1\) and \(Q(x) = x + 3\), their product is calculated as follows:

• Multiply: \((x^2 + 2x + 1) \cdot (x + 3)\)
• Distribute each term: \(x^2(x + 3) + 2x(x + 3) + 1(x + 3)\)
• Simplify: \(x^3 + 3x^2 + 2x^2 + 6x + x + 3\)
• Combine like terms: \(x^3 + 5x^2 + 7x + 3\)

The result is a new polynomial with a higher degree, demonstrating how polynomial multiplication works.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators (like addition and multiplication). They are the building blocks of algebraic operations and can represent real-world quantities that vary or change. Polynomial expressions are a subset of algebraic expressions, where only whole number exponents are used.

Algebraic expressions are structured using:

• Variables: Symbols (usually letters) that stand in for unknown or changing quantities.
• Coefficients: Numbers that multiply the variables in terms.
• Operators: Symbols that indicate mathematical operations (e.g., +, -, *).

For example, in the expression \(3x^2 - 4x + 5\):

• \(x\) is the variable.
• \(3\) and \(-4\) are coefficients.
• \(x^2\), \(-4x\), and \(+5\) are terms.

Understanding how to manipulate algebraic expressions, like by expanding or factoring them, is crucial in solving equations and modeling real-world scenarios.