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Leading Coefficient and Degree of Polynomial -9u⁴ + 12u⁸ - 12 + 5u
Mathematics
Middle School (Grade 7)
Question Content
What are the leading coefficient and degree of the polynomial? \n-9u⁴ + 12u⁸ - 12 + 5u
Correct Answer
Leading coefficient: 12; Degree: 8
Detailed Solution Steps
1
1. Identify the degree of each term: For a term \( au^n \), the degree is \( n \).
2
- \( -9u^4 \): degree = 4
3
- \( 12u^8 \): degree = 8
4
- \( -12 \) (or \( -12u^0 \)): degree = 0
5
- \( 5u \) (or \( 5u^1 \)): degree = 1
6
2. Determine the polynomial’s degree: The highest degree among all terms is 8 (from \( 12u^8 \)), so the polynomial’s degree is 8.
7
3. Find the leading coefficient: The term with the highest degree is \( 12u^8 \), so its coefficient (12) is the leading coefficient.
Knowledge Points Involved
1
Term of a Polynomial
A term in a polynomial is a monomial (e.g., \( -9u^4 \), \( 12u^8 \)) or constant (e.g., \( -12 \)), consisting of a coefficient and a variable raised to a non - negative integer power.
2
Degree of a Term
For a term \( au^n \) (non - zero \( a \), non - negative integer \( n \)), the degree is the exponent \( n \) of the variable. Constant terms (e.g., \( -12 \)) have degree 0 (since \( u^0 = 1 \)).
3
Degree of a Polynomial
The degree of a polynomial is the maximum degree of all its terms. It shows the highest power of the variable in the polynomial.
4
Leading Coefficient of a Polynomial
The leading coefficient is the coefficient of the term with the highest degree (when the polynomial is in descending degree order). It belongs to the 'leading term' (highest - degree term).
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