Problem 96 - Algebra and Trigonometry

Question

Make up several pairs of polynomials, then calculate the sum and product of each pair. On the basis of your experiments and observations, answer the following questions. (a) How is the degree of the product related to the degrees of the original polynomials? (b) How is the degree of the sum related to the degrees of the original polynomials? (c) Test your conclusions by finding the sum and product of the following polynomials: $$2 x^{3}+x-3 \quad \text { and } \quad-2 x^{3}-x+7$$

Step-by-Step Solution

Answer

Degree of product is sum of degrees; degree of sum is usually highest degree.

Step 1: Pair 1 - Select Polynomials

Select the first pair of polynomials as \( f(x) = 3x^2 + 4x + 1 \) and \( g(x) = 2x + 5 \).

Step 2: Pair 1 - Calculate Sum

The sum of polynomials \( f(x) = 3x^2 + 4x + 1 \) and \( g(x) = 2x + 5 \) is given by:\[ f(x) + g(x) = (3x^2 + 4x + 1) + (2x + 5) = 3x^2 + 6x + 6 \]The degree of the sum is 2, the highest degree from \( f(x) \).

Step 3: Pair 1 - Calculate Product

The product of \( f(x) = 3x^2 + 4x + 1 \) and \( g(x) = 2x + 5 \) is:\[ f(x) \cdot g(x) = (3x^2 + 4x + 1)(2x + 5) = 6x^3 + 15x^2 + 8x^2 + 20x + 2x + 5 \]Combine like terms to get:\[ 6x^3 + 23x^2 + 22x + 5 \]The degree of the product is 3, which is the sum of the degrees of \( f(x) \) and \( g(x) \).

Step 4: Pair 2 - Select Polynomials

Select the second pair of polynomials as \( f(x) = x^3 + 2x^2 - x + 3 \) and \( g(x) = x^4 + x^2 - x + 1 \).

Step 5: Pair 2 - Calculate Sum

The sum is:\[ (x^3 + 2x^2 - x + 3) + (x^4 + x^2 - x + 1) = x^4 + x^3 + 3x^2 - 2x + 4 \]The degree of the sum is 4, which is derived from the highest degree in either polynomial, that is \( g(x) \).

Step 6: Pair 2 - Calculate Product

The product is:\[ (x^3 + 2x^2 - x + 3)(x^4 + x^2 - x + 1) \]Calculate directly to get a polynomial of degree 7, which is the sum of the degrees (3 + 4).

Step 7: Observations from Experiments

**Observation (a):** The degree of the product equals the sum of the degrees of the individual polynomials. **Observation (b):** The degree of the sum equals the highest degree found in either polynomial.

Step 8: Given Polynomials - Calculate Sum

Given polynomials are \( 2x^3 + x - 3 \) and \( -2x^3 - x + 7 \).Their sum:\[ (2x^3 + x - 3) + (-2x^3 - x + 7) = 0x^3 + 0x + 4 = 4 \]The degree is 0, highlighting the output can have a lower degree when terms cancel each other.

Step 9: Given Polynomials - Calculate Product

The product:\[ (2x^3 + x - 3)(-2x^3 - x + 7) \]Calculating each term results in the highest degree being 6, as expected from the sum of degrees (3 + 3).

Step 10: Conclusion

**Conclusion (a):** The product's degree equals the sum of the degrees of original polynomials. **Conclusion (b):** The sum's degree is typically the highest amongst the original degrees, but can be lower if terms cancel out.

Key Concepts

Degree of Polynomials

Polynomials are mathematical expressions consisting of variables and coefficients. The degree of a polynomial is determined by the highest power of its variable. For example, in the polynomial \(3x^2 + 5x + 7\), the highest power of the variable \(x\) is 2, hence the degree is 2.

The degree is important because it gives insight into the polynomial's behavior and its graph's nature. Higher degree polynomials have more complex graphs with additional turning points. When performing arithmetic operations like addition, subtraction, or multiplication on polynomials, understanding their degree helps predict the outcome's degree.

• The degree of a sum of polynomials is no more than the highest degree found in the summands.
• The degree of the product of polynomials is the sum of the degrees of the individual factors.

Thus, comprehending a polynomial's degree is crucial for analyzing and manipulating polynomial expressions.

Sum of Polynomials

Adding polynomials involves combining like terms, which are terms having the same variable raised to the same power. Consider the polynomials \(f(x) = 3x^2 + 4x + 1\) and \(g(x) = 2x + 5\). Their sum is calculated by aligning them and adding corresponding coefficients:

\[f(x) + g(x) = (3x^2 + 4x + 1) + (2x + 5) = 3x^2 + 6x + 6\]

Notice that since there are no like terms with \(x^2\) in \(g(x)\), it simply remains as \(3x^2\).

• The degree of the sum is determined by the highest degree of the terms from any of the original polynomials.
• However, if terms of equal degree cancel each other out, as seen in \(2x^3 + x - 3\) and \(-2x^3 - x + 7\), the degree of the sum can be lower or even zero.

Mastering the sum of polynomials is a fundamental skill critical for algebraic proficiency.

Product of Polynomials

Multiplying polynomials involves applying the distributive property to every term in both expressions. Let's work through an example: multiplying \(f(x) = 3x^2 + 4x + 1\) with \(g(x) = 2x + 5\). Each term in \(f(x)\) is multiplied by each term in \(g(x)\):

\[ (3x^2 + 4x + 1) \cdot (2x + 5) = 3x^2 \cdot 2x + 3x^2 \cdot 5 + 4x \cdot 2x + 4x \cdot 5 + 1 \cdot 2x + 1 \cdot 5 \]Combine like terms to yield:
\[6x^3 + 15x^2 + 8x^2 + 20x + 2x + 5 = 6x^3 + 23x^2 + 22x + 5\]

From this process, the degree of the product is determined by adding the degrees of the highest terms in each polynomial (here \(2 + 1 = 3\)).

• This explains why multiplying results in a polynomial with a degree equal to the sum of the degrees of the factors.
• The complexity increases with higher-degree polynomials, leading to more cases of combining like terms for simplification.

Understanding polynomial multiplication paves the way for advanced algebra and calculus topics.