What are the leading coefficient and degree of the polynomia...

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--- ## Question 1 **Question content**: What are the leading coefficient and degree of the polynomial \(-9u^4 + 12u^8 - 12 + 5u\)? **Discipline**: Mathematics **Grade**: Middle School (Suitable for 7th - 9th grade, as it involves polynomial basics) ### Correct answer Leading coefficient: \( 12 \) Degree: \( 8 \) ### Detailed problem-solving steps 1. **Recall definitions**: - The **degree of a polynomial** is the highest exponent (power) of the variable in any term. - The **leading term** is the term with the highest degree, and the **leading coefficient** is the coefficient of this leading term. 2. **Identify degrees of all terms**: - For \(-9u^4\): The exponent of \( u \) is \( 4 \), so degree \( 4 \). - For \( 12u^8 \): The exponent of \( u \) is \( 8 \), so degree \( 8 \). - For \(-12\) (constant term): Can be written as \(-12u^0\), so degree \( 0 \). - For \( 5u \): Can be written as \( 5u^1 \), so degree \( 1 \). 3. **Determine the degree of the polynomial**: The highest degree among the terms is \( 8 \) (from \( 12u^8 \)). Thus, the degree of the polynomial is \( 8 \). 4. **Determine the leading coefficient**: The leading term is the term with the highest degree (\( 12u^8 \)). The coefficient of this term is \( 12 \), so the leading coefficient is \( 12 \). ### Knowledge points involved 1. **Degree of a polynomial** - Definition: The highest exponent of the variable in any term of the polynomial. For a term \( ax^n \) (where \( a \neq 0 \)), the degree is \( n \). Constant terms (e.g., \( -12 \)) have degree \( 0 \) (since \( -12 = -12x^0 \)). - Application: Used to classify polynomials (e.g., linear, quadratic, cubic) and compare their "complexity." 2. **Leading term and leading coefficient** - Leading term: The term with the highest degree in the polynomial. - Leading coefficient: The coefficient of the leading term. - Application: Helps in analyzing the end behavior of polynomials and simplifying polynomial expressions. 3. **Terms of a polynomial** - Definition: A polynomial is a sum of terms, where each term is a constant, a variable, or a product of a constant and a variable raised to a non-negative integer power (e.g., \( -9u^4 \), \( 12u^8 \), \( -12 \), \( 5u \) are terms). - Application: Essential for identifying components of a polynomial and performing operations (e.g., addition, subtraction) on polynomials. 4. **Standard form of a polynomial** - Definition: A polynomial written with terms in descending order of their degrees (e.g., \( 12u^8 - 9u^4 + 5u - 12 \) for the given polynomial). - Application: Simplifies identification of the leading term, degree, and leading coefficient. 5. **Exponents in polynomial terms** - Definition: The power of the variable in a term (e.g., in \( 12u^8 \), the exponent is \( 8 \)). - Application: Critical for determining the degree of individual terms and the polynomial as a whole. ---

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