Question

Let \(f\) be an \(n\) th-degree polynomial and let \(g\) be an \(m\) th-degree polynomial. What is the degree of the following polynomials? a. \(f \cdot f\) b. \(f^{\circ} f\) c. \(f \cdot g\) d. \(f^{\circ} g\)

Short answer

a. The product of a polynomial of degree n with itself. b. The composition of a polynomial of degree n with itself. c. The product of a polynomial of degree n with another polynomial of degree m. d. The composition of a polynomial of degree n with another polynomial of degree m. Answer: a. 2n b. n^2 c. n+m d. mn

Step-by-step solution

a. Degree of \(f \cdot f\)#

When a polynomial of degree \(n\) is multiplied by itself, the highest degree term in the result is obtained by multiplying the two highest degree terms. Since \(f\) is a polynomial of degree \(n\), its highest degree term is \(ax^n\). Thus, the highest degree term in the product will result from multiplying \(ax^n\) by itself: \((ax^n)(ax^n) = a^2x^{2n}\). The highest power of \(x\) in the product is \(2n\). So the degree of \(f \cdot f\) is \(2n\).

b. Degree of \(f^{\circ} f\)#

When a polynomial of degree \(n\) is composed with itself, the highest degree term in the composition is obtained by replacing every \(x\) in the original polynomial with the highest degree term of the polynomial to be composed. In our case, since \(f\) has degree \(n\), it will be replaced in every \(x\) of the original polynomial by \(ax^n\). Thus, the highest degree term in the composition will be \((ax^n)^n = a^nx^{n^2}\). So the degree of \(f^{\circ} f\) is \(n^2\).

c. Degree of \(f \cdot g\)#

When a polynomial \(f\) of degree \(n\) is multiplied by another polynomial \(g\) of degree \(m\), the highest degree term in the product is obtained by multiplying the highest degree terms of both polynomials. Let us consider the highest degree term of \(f\) to be \(ax^n\) and the highest degree term of \(g\) to be \(bx^m\). The highest degree term in the product will result from multiplying these two terms: \((ax^n)(bx^m) = abx^{n+m}\). So the degree of \(f \cdot g\) is \(n+m\).

d. Degree of \(f^{\circ} g\)#

When a polynomial \(f\) of degree \(n\) is composed with another polynomial \(g\) of degree \(m\), the highest degree term in the composition is obtained by replacing every \(x\) in the original polynomial with the highest degree term of the polynomial to be composed. In our case, since \(f\) has degree \(n\), it will be replaced in every \(x\) of the original polynomial by \(bx^m\). Thus, the highest degree term in the composition will be \((bx^m)^n = b^nx^{mn}\). So the degree of \(f^{\circ} g\) is \(mn\).

Key concepts

Degree Multiplication

Understanding the degree of multiplication in polynomials is essential when analyzing the behavior of equations in algebra. When two polynomials are multiplied, the degree of the resulting polynomial is determined by adding the degrees of the two polynomials being multiplied. This is because you are essentially multiplying the highest degree terms of each polynomial together.

For example, consider a polynomial \(f\) of degree \(n\) and another polynomial \(g\) of degree \(m\). If you multiply these, like \(f \cdot g\), the new polynomial created will have a degree of \(n + m\). When a polynomial is multiplied by itself, like \(f \cdot f\), the degree will double, resulting in a degree of \(2n\).

• Multiplying two polynomials involves combining their highest degrees.
• If both polynomials are identical, the degree doubles.

Degree Composition

Polynomial composition involves plugging one polynomial into another. Unlike multiplication, the degree of composition is a bit more nuanced. When composing, the degree of the resulting polynomial is a product of the degrees of the original polynomials.

Take a polynomial \(f\) of degree \(n\) and substitute another polynomial \(g\), also of degree \(m\), for every instance of \(x\) in \(f\). The resulting polynomial \(f^{\circ} g\) thereby has a degree of \(n \times m\).

• Composition means inserting one polynomial into another.
• The degrees multiply in composition, unlike addition in multiplication.

Polynomial Multiplication

Polynomial multiplication expands on degree multiplication, focusing on the actual process. Multiplying polynomials involves distributing each term of one polynomial with every term of the other. The degree of this resulting polynomial mirrors the degree rule: it’s the sum of the degrees of the two original polynomials.

For instance, multiplying \(f(x) = ax^n\) by \(g(x) = bx^m\) results in a highest degree term of \(abx^{n+m}\). Keeping the operations methodical ensures accurate results.

• Distribute every term of one polynomial across the other.
• Combine like terms for the final expression.
• The maximum degree term arises from the sum of the highest degrees.

Polynomial Composition

In polynomial composition, you replace variables in one polynomial with another polynomial. The intricacy mainly involves recalculating new terms based on substitution.

Consider replacing each \(x\) in \(f(x)\) with \(g(x)\), leading to a new polynomial that often has a degree equal to the product of the degrees of \(f\) and \(g\). A composition like \(f^{\circ} f\) (substituting \(f\) into itself) results in \(n^2\) degree, a unique perspective when compared to standard multiplication.

• Substitute a polynomial fully into another.
• The resulting degree reflects the multiplication of the original degrees.
• It's different from multiplication; here the degrees genuinely multiply.