Question

Determine the degree of the polynomial $$ 9 x^{2}(3 x-5)(5 x+1)^{4} $$

Short answer

The degree of the polynomial is 7.

Step-by-step solution

Identify the polynomial structure

The given expression is a product of multiple terms: \( 9x^2 \), \( (3x - 5) \), and \( (5x + 1)^4 \).

Determine the degree of each factor

Find the degree of each term individually: 1. The term \( 9x^2 \) has a degree of 2 because the highest power of \( x \) is 2. 2. The term \( 3x - 5 \) has a degree of 1 because the highest power of \( x \) is 1. 3. The term \( (5x + 1)^4 \) is a binomial raised to a power. Its degree is 4 times the degree of the binomial \( 5x + 1 \), which is 1. Therefore, the degree of \( (5x + 1)^4 \) is \( 4 \times 1 = 4 \).

Sum the degrees of each factor

Add the degrees of each term to find the overall degree of the polynomial: \( 2 + 1 + 4 = 7 \).

Key concepts

Polynomial Degree

When talking about the degree of a polynomial, we are interested in the highest power of the variable within an expression. For the polynomial in question, we decompose it into its factors to simplify the process of finding its degree.
This polynomial is given by the expression: \[9x^2 (3x - 5)(5x + 1)^4\]
Each term contributes to the overall degree.

The degree of a polynomial is a summation of the degrees of its individual parts when they are multiplied together.

Degree of a Term

The degree of a term is dictated by the exponent of the variable in that term. Let's break down each part of our polynomial:

• The term \[9x^2\] has a degree of 2. Here, the coefficient 9 doesn't affect the degree; only the exponent 2 on the variable x does.
• The term \[3x - 5\] has a degree of 1 because the variable x is raised to the first power. The constant -5 does not change the degree.
• The term \[ (5x + 1)^4\] is a bit different. It's a binomial raised to the fourth power. Inside the parenthesis, the highest power of x is 1. Therefore, when raised to the power of 4, the degree of this term becomes \[ 4 \times 1 = 4\].

Binomial Theorem

The binomial theorem is a key concept when dealing with expressions like \[ (5x + 1)^4\]. While it is used for expanding binomials raised to a power, it also helps in understanding how the degree works. The theorem states that:\[ (a + b)^n \ = \ \sum_{k=0}^n C(n,k) a^{n-k} b^k \]This complex formula is succinctly useful for finding degrees. In our case, knowing that \[ (5x + 1)^4\] represents a polynomial where the highest power is \[ 4 \times 1 = 4\].

Polynomial Multiplication

When multiplying polynomials, the total degree of the resulting polynomial is the sum of the degrees of the multiplied factors. In our example, we multiply \[ 9x^2\], \[ 3x - 5\], and \[ (5x + 1)^4\]:

• Degree of \[ 9x^2\]: 2
• Degree of \[ 3x - 5\]: 1
• Degree of \[ (5x + 1)^4\]: 4

Adding these degrees together, we get \[ 2 + 1 + 4 = 7\]. Thus, the degree of the polynomial \[ 9x^2 (3x - 5) (5x + 1)^4\] is 7. This method shows the importance of knowing how to handle each term independently before combining them.