Problem 61 - Precalculus with Calculus Previews

Question

Suppose \(f\) is a polynomial function of degree \(m\) and \(g\) is a polynomial function of degree \(n\). What is the degree of \(f+g\) ? Of \(f g\) ?

Step-by-Step Solution

Answer

Degree of \( f+g \) is max(\( m, n \)), and degree of \( fg \) is \( m+n \).

Step 1: Analyzing Degrees of Polynomials

Recall that a polynomial of degree \( m \) has the form \( a_m x^m + a_{m-1} x^{m-1} + \, \cdots \, + a_0 \), where \( a_m eq 0 \). Similarly, a polynomial of degree \( n \) is \( b_n x^n + b_{n-1} x^{n-1} + \, \cdots \, + b_0 \), where \( b_n eq 0 \).

Step 2: Determining Degree of Sum (f + g)

The degree of the sum of two polynomials is the highest degree of the terms in the resulting polynomial. Therefore, if \( m > n \), the degree of \( f+g \) is \( m \). If \( n > m \), it is \( n \). If \( m = n \), the degree of \( f+g \) is still \( m \) or \( n \), unless the leading coefficients cancel each other out, in which case it might be less.

Step 3: Determining Degree of Product (f * g)

The degree of the product of two polynomials is the sum of their degrees. Therefore, the degree of \( f \cdot g \) is \( m + n \). This comes from multiplying the highest degree terms in both polynomials, which gives the term \( a_m \, b_n \, x^{m+n} \).

Key Concepts

Degree of Polynomials

In the world of polynomial functions, understanding the degree is fundamental. The degree of a polynomial is defined as the highest power of the variable x in the expression. For example, in the polynomial expression \( a_m x^m + a_{m-1} x^{m-1} + \cdots + a_0 \), the degree is \( m \). This is because \( x^m \) is the term with the highest exponent in the polynomial.
The degree gives us valuable information about the polynomial's shape and the number of roots it can have. The highest power tells us how many times the graph can intersect the x-axis. For instance:

• A linear polynomial (degree 1) looks like a straight line and intersects the x-axis once.
• A quadratic polynomial (degree 2) forms a parabola and can intersect the x-axis twice.
• A cubic polynomial (degree 3) has a wavy shape and can intersect the x-axis up to three times.

Recognizing the degree of polynomials helps in simplifying calculations, predicting graph behavior, and understanding possible solutions or roots.

Polynomial Addition

Adding polynomials involves combining like terms, terms with the same power of \( x \). When you have two polynomials \( f(x) \) and \( g(x) \), and you add them to form \( f+g \), the degree of \( f+g \) is determined by the highest degree term present in the result.
It's important to note some possible outcomes:

• If the degree of \( f \) is greater than that of \( g \), then the degree of \( f+g \) is simply the degree of \( f \).
• If the degree of \( g \) is greater than that of \( f \), the degree of \( f+g \) is the degree of \( g \).
• If both \( f \) and \( g \) have the same degree, typically the result maintains that degree, unless the leading coefficients cancel each other out, potentially lowering the degree.

Always watch out for terms that may nullify each other, especially when adding polynomials with similar leading terms.

Polynomial Multiplication

When multiplying two polynomials, you essentially distribute each term in the first polynomial across every term in the second polynomial. This process will result in new terms where the degrees of terms from each polynomial are added together.
To find the degree of the product \( f \cdot g \), sum the degrees of the individual polynomials. So, if \( f \) is of degree \( m \) and \( g \) is of degree \( n \), then \( f \cdot g \) will have a degree of \( m + n \).

• This is because when you take the highest degree term from \( f \) (i.e., \( a_m x^m \)) and multiply it by the highest degree term from \( g \) (i.e., \( b_n x^n \)), you get \( a_m b_n x^{m+n} \), which defines the degree of the resulting polynomial.
• Polynomial multiplication is precise in determining the maximum number of zeros, which is related to the degree, as it provides the sum of the zeros for both factors.

Mastering polynomial multiplication allows more complex expressions to be broken down into simpler parts, making problem-solving more manageable and understandable.