Let f(x) and g(x) be two polynomials (with real coefficients) having degree 3 and 4 respectively. What is the degree of f(x) g(x)
7
Understanding Polynomials and Their Degrees
A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. The degree of a polynomial is the highest exponent of the variable in the expression.
For example:
- \( P(x) = 3x^2 + 5x - 1 \) is a polynomial of degree 2.
- \( Q(x) = 7x^4 - 2x^3 + x^2 + 9 \) is a polynomial of degree 4.
Finding the Degree of the Product of Two Polynomials
When we multiply two non-zero polynomials, the degree of the resulting polynomial is found by adding the degrees of the individual polynomials. Let's say we have two polynomials, \( f(x) \) and \( g(x) \).
- Let \( \text{deg}(f(x)) \) represent the degree of polynomial \( f(x) \).
- Let \( \text{deg}(g(x)) \) represent the degree of polynomial \( g(x) \).
The degree of the product \( f(x) \cdot g(x) \) is given by the formula:
\( \text{deg}(f(x) \cdot g(x)) = \text{deg}(f(x)) + \text{deg}(g(x)) \)
Applying the Degree Rule to f(x) and g(x)
In this specific problem, we are given two polynomials:
- \( f(x) \) is a polynomial with degree 3. So, \( \text{deg}(f(x)) = 3 \).
- \( g(x) \) is a polynomial with degree 4. So, \( \text{deg}(g(x)) = 4 \).
To find the degree of the product \( f(x) \cdot g(x) \), we use the rule:
\( \text{deg}(f(x) \cdot g(x)) = \text{deg}(f(x)) + \text{deg}(g(x)) \)
Substituting the given degrees:
\( \text{deg}(f(x) \cdot g(x)) = 3 + 4 \)
\( \text{deg}(f(x) \cdot g(x)) = 7 \)
Thus, the degree of the product of the two polynomials \( f(x) \) and \( g(x) \) is 7.
Analyzing the Options
Let's look at the given options based on our calculation:
| Option | Value | Analysis |
|---|---|---|
| 1 | 12 | This would be the case if we multiplied the degrees (3 * 4 = 12), which is incorrect for finding the degree of a polynomial product. |
| 2 | 7 | This matches our calculation (3 + 4 = 7). This is the correct degree for the product of these polynomials. |
| 3 | 4 | This is the degree of g(x), not the product. |
| 4 | 3 | This is the degree of f(x), not the product. |
Based on the rule for the degree of the product of polynomials, the correct degree is 7.
Revision Table: Polynomial Degree Properties
| Operation | Rule for Degree of Result (P and Q are non-zero polynomials) | Example |
|---|---|---|
| Addition/Subtraction: \( P(x) \pm Q(x) \) | \( \text{deg}(P \pm Q) \le \max(\text{deg}(P), \text{deg}(Q)) \). If \( \text{deg}(P) \ne \text{deg}(Q) \), then \( \text{deg}(P \pm Q) = \max(\text{deg}(P), \text{deg}(Q)) \). If \( \text{deg}(P) = \text{deg}(Q) \), the degree can be less than or equal to this maximum (if leading terms cancel). | \( (x^3+x) + (x^2+1) \): degree 3. \( (x^3+x) - (x^3+1) \): degree 1. |
| Multiplication: \( P(x) \cdot Q(x) \) | \( \text{deg}(P \cdot Q) = \text{deg}(P) + \text{deg}(Q) \) | \( (x^3+1) \cdot (x^2+1) = x^5+x^3+x^2+1 \): degree 5 (3+2). |
| Division: \( P(x) / Q(x) \) | \( \text{deg}(P / Q) = \text{deg}(P) - \text{deg}(Q) \), for the quotient polynomial, assuming \( \text{deg}(P) \ge \text{deg}(Q) \). | \( (x^5+x^3+x^2+1) / (x^2+1) = x^3+1 \): degree 3 (5-2). |
| Composition: \( P(Q(x)) \) | \( \text{deg}(P(Q(x))) = \text{deg}(P) \cdot \text{deg}(Q) \) | \( P(x)=x^2, Q(x)=x^3 \). \( P(Q(x)) = P(x^3) = (x^3)^2 = x^6 \): degree 6 (2*3). |
Additional Information on Polynomial Degrees
The degree of a polynomial is a fundamental property that tells us a lot about its behavior, such as the maximum number of roots it can have (Fundamental Theorem of Algebra). Understanding how degrees change under different operations is crucial in algebra.
- The product rule for degrees holds true for any non-zero polynomials with real or complex coefficients.
- If one of the polynomials is the zero polynomial (which has an undefined degree or sometimes degree -1), the product is the zero polynomial, and its degree is handled specially. However, the question specifies polynomials with defined degrees (3 and 4), implying they are non-zero.
- When multiplying polynomials, the term with the highest degree is obtained by multiplying the terms with the highest degrees from each polynomial. For \( f(x) \) with leading term \( ax^3 \) (\( a \ne 0 \)) and \( g(x) \) with leading term \( bx^4 \) (\( b \ne 0 \)), the leading term of \( f(x)g(x) \) will be \( (ax^3)(bx^4) = abx^{3+4} = abx^7 \). Since \( a \ne 0 \) and \( b \ne 0 \), \( ab \ne 0 \), so the highest power of \( x \) is \( x^7 \), giving a degree of 7.
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