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Chapter 5: Problem 20

Determine the coefficient and the degree of each term in each polynomial. \(9 a^{3}-4 a^{2}\)

Short Answer

Expert verified
The coefficients are 9 and -4. The degrees are 3 and 2.

Step by step solution

01

Identify the individual terms

The polynomial given is (9 a^{3} - 4 a^{2}). The individual terms are (9 a^{3}) and (-4 a^{2}).
02

Determine the coefficient of the first term

The coefficient of the first term, (9 a^{3}), is (9).
03

Determine the degree of the first term

The degree of the first term, (9 a^{3}), is (3) because the exponent of (a) is (3).
04

Determine the coefficient of the second term

The coefficient of the second term, (-4 a^{2}), is (-4).
05

Determine the degree of the second term

The degree of the second term, (-4 a^{2}), is (2) because the exponent of (a) is (2).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coefficient
In a polynomial, the coefficient is the numerical factor of a term. It is the number that is directly in front of the variable. In our example, the polynomial is given as \(9 a^{3}-4 a^{2}\).Look at each term, starting with \(9 a^{3}\), the coefficient here is 9. It's simply the number multiplying the variable part of the term.
The second term is \(-4 a^{2}\), and here, the coefficient is -4. Notice that the sign in front of the number is part of the coefficient. So if the term is negative, include the negative sign when identifying the coefficient.
Degree of a Term
The degree of a term is the exponent of the variable in that term. It tells you the power to which the variable is raised. In the term, \(9 a^{3}\), the degree is 3 because the variable \(a\) is raised to the power of 3.
Moving on to the second term, \(-4 a^{2}\), here the degree is 2. Again, this is because the variable \(a\) is raised to the power of 2. The degree indicates the highest power of the variable within that particular term.
Exponent
An exponent refers to how many times a number (the base) is multiplied by itself. It is expressed as a small number placed to the upper right of the base. In the polynomial \(9 a^{3}-4 a^{2}\), look at the term \(9 a^{3}\). Here, the base is \(a\) and the exponent is 3. This means \(a\) is multiplied by itself three times: \(a \times a \times a\).
For the term \(-4 a^{2}\), the base is again \(a\) and the exponent is 2. This tells us that \(a\) is multiplied by itself twice: \(a \times a\).
Individual Terms
Polynomials are made up of individual terms, each of which includes a coefficient and a variable raised to an exponent. In the polynomial \(9 a^{3}-4 a^{2}\), identifying the individual terms helps break down and understand the components of the expression. Here, the polynomial consists of two individual terms: \(9 a^{3}\), and \(-4 a^{2}\).
Each term is a mix of a number (coefficient), a variable, and an exponent. By examining each term separately, you can easily determine their coefficients and degrees.

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