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Sagot :
Sure! Let's work through the problem step-by-step focusing on what you need to fill in the table.
1. Understanding columns and terminology:
- Monomial: A monomial is an algebraic expression consisting of one term. It can include constants, variables, and exponents, but must not contain any addition or subtraction operators.
- Coefficient: The coefficient in a term is the numerical factor multiplied by the variables.
- Literal Part: The literal part of a monomial refers to all the variables and their powers in the term.
- Degree: The degree of a monomial is the sum of the exponents of all the variables in the term.
Let's now analyze each given monomial and fill out the table.
1. For [tex]$-2k^3$[/tex]:
- Coefficient: The numerical factor is [tex]\(-2\)[/tex].
- Literal Part: The variables and their exponents are [tex]\(k^3\)[/tex].
- Degree: Since the only variable is [tex]\(k\)[/tex], and its exponent is [tex]\(3\)[/tex], the degree is [tex]\(3\)[/tex].
[tex]\[ -2k^3 \quad \longrightarrow \quad \text{Coefficient: } -2, \text{ Literal Part: } k^3, \text{ Degree: } 3 \][/tex]
2. For [tex]\(213yz\)[/tex]:
- Coefficient: The numerical factor is [tex]\(213\)[/tex].
- Literal Part: The variables and their exponents are [tex]\(yz\)[/tex].
- Degree: The exponents are [tex]\(1\)[/tex] for both [tex]\(y\)[/tex] and [tex]\(z\)[/tex], so the degree is [tex]\(1 + 1 = 2\)[/tex].
[tex]\[ 213yz \quad \longrightarrow \quad \text{Coefficient: } 213, \text{ Literal Part: } yz, \text{ Degree: } 2 \][/tex]
3. For [tex]\(+\frac{7}{3} y^3\)[/tex]:
- Coefficient: The numerical factor is [tex]\(\frac{7}{3}\)[/tex].
- Literal Part: The variables and their exponents are [tex]\(y^3\)[/tex].
- Degree: The only variable [tex]\(y\)[/tex] has an exponent of [tex]\(3\)[/tex], so the degree is [tex]\(3\)[/tex].
[tex]\[ +\frac{7}{3} y^3 \quad \longrightarrow \quad \text{Coefficient: } \frac{7}{3}, \text{ Literal Part: } y^3, \text{ Degree: } 3 \][/tex]
4. For [tex]\(\sqrt{2k^x}\)[/tex]:
- Coefficient: The numerical factor is [tex]\(\sqrt{2}\)[/tex].
- Literal Part: The variables and their exponents are [tex]\(k^x\)[/tex].
- Degree: If the variable inside the square root has an exponent [tex]\(x\)[/tex], then the degree is [tex]\(x\)[/tex].
[tex]\[ \sqrt{2k^x} \quad \longrightarrow \quad \text{Coefficient: } \sqrt{2}, \text{ Literal Part: } k^x, \text{ Degree: } x \][/tex]
5. For the entry with only [tex]\(-3\)[/tex]:
- Coefficient: The numerical factor is [tex]\(-3\)[/tex].
- Literal Part: The literal part given is [tex]\(xy^3\)[/tex].
- Degree: Since [tex]\(x\)[/tex] has an implied exponent of [tex]\(1\)[/tex] and [tex]\(y\)[/tex] has an exponent of [tex]\(3\)[/tex], the total degree is [tex]\(1 + 3 = 4\)[/tex].
[tex]\[ -3xy^3 \quad \longrightarrow \quad \text{Coefficient: } -3, \text{ Literal Part: } xy^3, \text{ Degree: } 4 \][/tex]
6. For [tex]\(\frac{W}{7}\)[/tex]:
- Coefficient: The numerical factor is [tex]\(\frac{1}{7}\)[/tex] since [tex]\(W\)[/tex] is treated as the variable.
- Literal Part: The variable is [tex]\(W\)[/tex].
- Degree: The variable [tex]\(W\)[/tex] has an implied exponent of [tex]\(1\)[/tex], so the degree is [tex]\(1\)[/tex].
[tex]\[ \frac{W}{7} \quad \longrightarrow \quad \text{Coefficient: } \frac{1}{7}, \text{ Literal Part: } W, \text{ Degree: } 1 \][/tex]
7. For [tex]\(-\frac{6mm}{11}\)[/tex]:
- Coefficient: The numerical factor is [tex]\(-\frac{6}{11}\)[/tex].
- Literal Part: The variables and their exponents are [tex]\(m^2\)[/tex] (since [tex]\(mm = m^2\)[/tex]).
- Degree: The variable [tex]\(m\)[/tex] has an exponent of [tex]\(2\)[/tex], so the degree is [tex]\(2\)[/tex].
[tex]\[ -\frac{6mm}{11} \quad \longrightarrow \quad \text{Coefficient: } -\frac{6}{11}, \text{ Literal Part: } m^2, \text{ Degree: } 2 \][/tex]
Thus, the completed table looks like:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline Monômio & Coeficiente & Parte Literal & Grau \\ \hline $-2 k^3$ & $-2$ & $k^3$ & $3$ \\ \hline $213yz$ & $213$ & $yz$ & $2$ \\ \hline $+\frac{7}{3} y^3$ & $\frac{7}{3}$ & $y^3$ & $3$ \\ \hline $\sqrt{2 k^x}$ & $\sqrt{2}$ & $k^x$ & $x$ \\ \hline & $-3$ & $xy^3$ & $4$ \\ \hline $\frac{W}{7}$ & $\frac{1}{7}$ & $W$ & $1$ \\ \hline $-\frac{6 m^2}{11}$ & $-\frac{6}{11}$ & $m^2$ & $2$ \\ \hline \end{tabular} \][/tex]
1. Understanding columns and terminology:
- Monomial: A monomial is an algebraic expression consisting of one term. It can include constants, variables, and exponents, but must not contain any addition or subtraction operators.
- Coefficient: The coefficient in a term is the numerical factor multiplied by the variables.
- Literal Part: The literal part of a monomial refers to all the variables and their powers in the term.
- Degree: The degree of a monomial is the sum of the exponents of all the variables in the term.
Let's now analyze each given monomial and fill out the table.
1. For [tex]$-2k^3$[/tex]:
- Coefficient: The numerical factor is [tex]\(-2\)[/tex].
- Literal Part: The variables and their exponents are [tex]\(k^3\)[/tex].
- Degree: Since the only variable is [tex]\(k\)[/tex], and its exponent is [tex]\(3\)[/tex], the degree is [tex]\(3\)[/tex].
[tex]\[ -2k^3 \quad \longrightarrow \quad \text{Coefficient: } -2, \text{ Literal Part: } k^3, \text{ Degree: } 3 \][/tex]
2. For [tex]\(213yz\)[/tex]:
- Coefficient: The numerical factor is [tex]\(213\)[/tex].
- Literal Part: The variables and their exponents are [tex]\(yz\)[/tex].
- Degree: The exponents are [tex]\(1\)[/tex] for both [tex]\(y\)[/tex] and [tex]\(z\)[/tex], so the degree is [tex]\(1 + 1 = 2\)[/tex].
[tex]\[ 213yz \quad \longrightarrow \quad \text{Coefficient: } 213, \text{ Literal Part: } yz, \text{ Degree: } 2 \][/tex]
3. For [tex]\(+\frac{7}{3} y^3\)[/tex]:
- Coefficient: The numerical factor is [tex]\(\frac{7}{3}\)[/tex].
- Literal Part: The variables and their exponents are [tex]\(y^3\)[/tex].
- Degree: The only variable [tex]\(y\)[/tex] has an exponent of [tex]\(3\)[/tex], so the degree is [tex]\(3\)[/tex].
[tex]\[ +\frac{7}{3} y^3 \quad \longrightarrow \quad \text{Coefficient: } \frac{7}{3}, \text{ Literal Part: } y^3, \text{ Degree: } 3 \][/tex]
4. For [tex]\(\sqrt{2k^x}\)[/tex]:
- Coefficient: The numerical factor is [tex]\(\sqrt{2}\)[/tex].
- Literal Part: The variables and their exponents are [tex]\(k^x\)[/tex].
- Degree: If the variable inside the square root has an exponent [tex]\(x\)[/tex], then the degree is [tex]\(x\)[/tex].
[tex]\[ \sqrt{2k^x} \quad \longrightarrow \quad \text{Coefficient: } \sqrt{2}, \text{ Literal Part: } k^x, \text{ Degree: } x \][/tex]
5. For the entry with only [tex]\(-3\)[/tex]:
- Coefficient: The numerical factor is [tex]\(-3\)[/tex].
- Literal Part: The literal part given is [tex]\(xy^3\)[/tex].
- Degree: Since [tex]\(x\)[/tex] has an implied exponent of [tex]\(1\)[/tex] and [tex]\(y\)[/tex] has an exponent of [tex]\(3\)[/tex], the total degree is [tex]\(1 + 3 = 4\)[/tex].
[tex]\[ -3xy^3 \quad \longrightarrow \quad \text{Coefficient: } -3, \text{ Literal Part: } xy^3, \text{ Degree: } 4 \][/tex]
6. For [tex]\(\frac{W}{7}\)[/tex]:
- Coefficient: The numerical factor is [tex]\(\frac{1}{7}\)[/tex] since [tex]\(W\)[/tex] is treated as the variable.
- Literal Part: The variable is [tex]\(W\)[/tex].
- Degree: The variable [tex]\(W\)[/tex] has an implied exponent of [tex]\(1\)[/tex], so the degree is [tex]\(1\)[/tex].
[tex]\[ \frac{W}{7} \quad \longrightarrow \quad \text{Coefficient: } \frac{1}{7}, \text{ Literal Part: } W, \text{ Degree: } 1 \][/tex]
7. For [tex]\(-\frac{6mm}{11}\)[/tex]:
- Coefficient: The numerical factor is [tex]\(-\frac{6}{11}\)[/tex].
- Literal Part: The variables and their exponents are [tex]\(m^2\)[/tex] (since [tex]\(mm = m^2\)[/tex]).
- Degree: The variable [tex]\(m\)[/tex] has an exponent of [tex]\(2\)[/tex], so the degree is [tex]\(2\)[/tex].
[tex]\[ -\frac{6mm}{11} \quad \longrightarrow \quad \text{Coefficient: } -\frac{6}{11}, \text{ Literal Part: } m^2, \text{ Degree: } 2 \][/tex]
Thus, the completed table looks like:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline Monômio & Coeficiente & Parte Literal & Grau \\ \hline $-2 k^3$ & $-2$ & $k^3$ & $3$ \\ \hline $213yz$ & $213$ & $yz$ & $2$ \\ \hline $+\frac{7}{3} y^3$ & $\frac{7}{3}$ & $y^3$ & $3$ \\ \hline $\sqrt{2 k^x}$ & $\sqrt{2}$ & $k^x$ & $x$ \\ \hline & $-3$ & $xy^3$ & $4$ \\ \hline $\frac{W}{7}$ & $\frac{1}{7}$ & $W$ & $1$ \\ \hline $-\frac{6 m^2}{11}$ & $-\frac{6}{11}$ & $m^2$ & $2$ \\ \hline \end{tabular} \][/tex]
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