Problem 28
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. $$ x^{4} y^{3} z^{2}+9 z $$
Step-by-Step Solution
Verified Answer
Answer: The given polynomial is a binomial, has a degree of 9, and the numerical coefficients are 1 and 9 for the respective terms.
1Step 1: Identify the polynomial type
To figure out if the given polynomial is a monomial, binomial, or trinomial, count the number of terms in the expression. In this case, we have:
$$
x^{4}y^{3}z^{2} + 9z
$$
There are two terms: \(x^{4}y^{3}z^{2}\) and \(9z\). So, the given polynomial is a binomial.
2Step 2: Determine the degree of the polynomial
To find the degree of the polynomial, we have to find the highest power of any variable in the expression, including the sum of the powers if there are multiple variables in a single term. Here, we have:
1. For the first term, \(x^{4}y^{3}z^{2}\), the sum of the powers is \(4+3+2=9\).
2. For the second term, \(9z\), the power of the variable \(z\) is \(1\).
Since term \(x^{4}y^{3}z^{2}\) has the highest degree (9), we can say that the degree of the polynomial is 9.
3Step 3: Find the numerical coefficients
The numerical coefficient is the number in front of each term. In this case, we have:
1. For the first term, \(x^{4}y^{3}z^{2}\), the numerical coefficient is 1 (since there is no explicit number in front of the term, we assume it to be 1).
2. For the second term, \(9z\), the numerical coefficient is 9.
Therefore, the numerical coefficients of the terms are 1 and 9, respectively.
Key Concepts
MonomialBinomialTrinomialDegree of a PolynomialNumerical Coefficient
Monomial
A monomial is an algebraic expression that consists of a single term. It typically includes a variable (or variables) raised to an exponent, and can also include a numerical coefficient, although the coefficient is not always explicitly written.
For example, the term \(7x^3\) is a monomial. It has a numerical coefficient of 7 and a variable \(x\) raised to the power of 3. If you see something like \(x^4\), this is also a monomial with an implied numerical coefficient of 1 since any number multiplied by 1 is itself. The key characteristic of a monomial is that it has no addition or subtraction signs that combine it with other terms.
For example, the term \(7x^3\) is a monomial. It has a numerical coefficient of 7 and a variable \(x\) raised to the power of 3. If you see something like \(x^4\), this is also a monomial with an implied numerical coefficient of 1 since any number multiplied by 1 is itself. The key characteristic of a monomial is that it has no addition or subtraction signs that combine it with other terms.
Binomial
A binomial, as the prefix 'bi-' suggests, is a polynomial with exactly two terms. These terms are usually separated by a plus (+) or a minus (−) sign.
Consider the expression \(3x^2 + 4y\). This is a classic example of a binomial: it has two distinct terms, \(3x^2\) and \(4y\), and each term has its own distinct variables and exponents. In the original exercise, the polynomial \(x^{4} y^{3} z^{2} + 9 z\) is identified as a binomial because it consists of exactly two terms that are not like terms and therefore cannot be combined.
Consider the expression \(3x^2 + 4y\). This is a classic example of a binomial: it has two distinct terms, \(3x^2\) and \(4y\), and each term has its own distinct variables and exponents. In the original exercise, the polynomial \(x^{4} y^{3} z^{2} + 9 z\) is identified as a binomial because it consists of exactly two terms that are not like terms and therefore cannot be combined.
Trinomial
A trinomial is an algebraic expression made up of three terms connected by addition or subtraction signs.
For example, the expression \(2x^2 - 3x + 4\) is a trinomial. It contains three terms: \(2x^2\), \(-3x\), and \(4\). Each term may contain constants, coefficients, variables, and exponents, but together they form a single three-part expression. The trinomial resembles a binomial, but with one extra term.
For example, the expression \(2x^2 - 3x + 4\) is a trinomial. It contains three terms: \(2x^2\), \(-3x\), and \(4\). Each term may contain constants, coefficients, variables, and exponents, but together they form a single three-part expression. The trinomial resembles a binomial, but with one extra term.
Degree of a Polynomial
The degree of a polynomial is determined by the term with the highest sum of exponents on its variables. It's a very important characteristic as it provides valuable information about the behavior of the polynomial, especially concerning its graph.
For a given term, if there's just one variable, like \(x^5\), then the degree is simply 5, the exponent. However, if the term has multiple variables, like \(x^4y^3z^2\), then we add up all the exponents: 4 + 3 + 2, to get a degree of 9. The degree of a polynomial is the highest degree among all its terms, so in the case where the highest degree term is \(x^4y^3z^2\), the polynomial itself is said to have a degree of 9.
For a given term, if there's just one variable, like \(x^5\), then the degree is simply 5, the exponent. However, if the term has multiple variables, like \(x^4y^3z^2\), then we add up all the exponents: 4 + 3 + 2, to get a degree of 9. The degree of a polynomial is the highest degree among all its terms, so in the case where the highest degree term is \(x^4y^3z^2\), the polynomial itself is said to have a degree of 9.
Numerical Coefficient
The numerical coefficient in a polynomial term is the numerical factor that is multiplied by the variable or variables. It's the number that 'stands in front' of the term. The coefficient provides information on the term’s scale within a polynomial.
For instance, in the term \(8x^2\), 8 is the numerical coefficient. If a term does not have an explicit numerical coefficient written, such as \(y^3\), it is understood to have a coefficient of 1, since any number multiplied by 1 remains unchanged. Identifying the numerical coefficient is crucial when solving equations as it affects both sides of an equation and plays a significant role in the graphing of polynomial functions.
For instance, in the term \(8x^2\), 8 is the numerical coefficient. If a term does not have an explicit numerical coefficient written, such as \(y^3\), it is understood to have a coefficient of 1, since any number multiplied by 1 remains unchanged. Identifying the numerical coefficient is crucial when solving equations as it affects both sides of an equation and plays a significant role in the graphing of polynomial functions.
Other exercises in this chapter
Problem 28
For the following problems, find the products. $$ (5 a-6)^{2} $$
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For the following problems, simplify each of the algebraic expressions. $$ 3 a^{2}+6 a^{2}+2 a^{2} $$
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Perform the following additions and subtractions. Subtract \(4 s-3\) from \(7 s+8\)
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Use numerical evaluation on the equations. Geometry (perimeter of a rectangle) \(P=2 l+2 w . \) Find \(P\) if \(l=8 \frac{1}{4}\) and \(w=12 \frac{8}{9}\).
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