Problem 31 - Elementary Algebra

Question

For the following problems, classify each polynomial as a monomial, binomial, or trinomial. State the degree of each polynomial and write the numerical coefficient of each term. $$ -9+3 x^{2}+2 x y 6 z^{2} $$

Step-by-Step Solution

Answer

Question: Given the polynomial $$-9+3 x^{2}+2 x y 6 z^{2}$$, classify the polynomial, state its degree, and list the numerical coefficients of each term. Answer: The polynomial is a trinomial with a degree of 4. The numerical coefficients of the terms are -9, 3, and 12.

Step 1: Classify the Polynomial

First, let's count the number of terms in the polynomial. There are 3 terms: 1. $$-9$$ 2. $$3 x^{2}$$ 3. $$2 x y 6 z^{2}$$ Since there are 3 terms, this is a trinomial.

Step 2: State the Degree of the Polynomial

Now, let's find the degree of each term: 1. Term 1: $$-9$$ has a degree of 0 as it is a constant term. 2. Term 2: $$3 x^{2}$$ has a degree of 2, as the highest power of the variable x is 2. 3. Term 3: $$2 x y 6 z^{2}$$ has a degree of 1+1+2=4, as the sum of the powers of each variable is 4. The degree of a polynomial is the highest degree among its terms, in this case, it's 4.

Step 3: Determine the Numerical Coefficients for Each Term

Finally, let's determine the numerical coefficients of each term: 1. Term 1: $$-9$$ has a numerical coefficient of -9. 2. Term 2: $$3 x^{2}$$ has a numerical coefficient of 3. 3. Term 3: $$2 x y 6 z^{2}$$ has a numerical coefficient of 2*6=12. So, we have classified the polynomial as a trinomial, determined its degree to be 4, and found the numerical coefficients for each term: -9, 3, and 12.

Key Concepts

Trinomial

A polynomial is an expression that consists of terms, where each term is made up of numbers, variables, and exponents. The number of terms directly influences how we classify polynomials. Three core classifications are:

• Monomial: A single term.
• Binomial: Two terms.
• Trinomial: Three terms.

For instance, when you encounter the expression \[-9 + 3x^2 + 2xy6z^2\], you'll notice that it contains three distinct terms: \(-9\), \(3x^2\), and \(2xy6z^2\). Therefore, this expression falls under the trinomial category. Remember, noting the number of terms and their variety can help categorize polynomials easily.

Degree of Polynomial

The degree of a polynomial is a key concept. It tells us about the highest power of any variable present in any of the terms. This degree can help in understanding how functions behave at their extremities. To determine the degree, focus on each term separately and identify the highest power:

• For a constant term like \(-9\), the degree is 0, as it has no variable part.
• In \(3x^2\), the degree is 2, which comes from the exponent of \(x^2\).
• With \(2xy6z^2\), you find the degree by adding the powers of each variable; \(x^1, y^1, z^2\) sum up to give a degree of 4.

Thus, the degree of the polynomial is the highest degree from these terms, which is 4 in this case. Understanding this helps in predicting polynomial function behavior and represents how quickly the values can grow or shrink.

Numerical Coefficient

Numerical coefficients form the backbone of each term in a polynomial; they are the constant multipliers that stand with the variable components. Identifying these helps simplify, combine, and perform other algebraic operations:

• In the term \(-9\), the numerical coefficient is \(-9\) itself.
• For \(3x^2\), the numerical coefficient is \(3\).
• In \(2xy6z^2\), we multiply the numbers \(2\) and \(6\) to get a numerical coefficient of \(12\).

By focusing solely on these coefficients, you can often simplify expressions and even solve equations more effectively. Knowing how to correctly identify them is a critical skill across many branches of algebra.