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Find the Leading Coefficient and Degree of the Polynomial 23y² - 9y³ - 9y + 2
Mathematics
Grade 9 (Junior High School)
Question Content
What are the leading coefficient and degree of the polynomial $23y^2 - 9y^3 - 9y + 2$?
Correct Answer
Leading coefficient: -9; Degree: 3
Detailed Solution Steps
1
Step 1: Rewrite the polynomial in standard form, which means arranging the terms in descending order of the exponent of the variable. For $23y^2 - 9y^3 - 9y + 2$, the standard form is $-9y^3 + 23y^2 - 9y + 2$.
2
Step 2: Identify the leading coefficient: this is the coefficient of the term with the highest exponent in the standard form of the polynomial. In $-9y^3 + 23y^2 - 9y + 2$, the coefficient of the highest-degree term $-9y^3$ is -9, so the leading coefficient is -9.
3
Step 3: Identify the degree of the polynomial: this is the highest exponent value of the variable in the polynomial. The highest exponent in $-9y^3 + 23y^2 - 9y + 2$ is 3, so the degree of the polynomial is 3.
Knowledge Points Involved
1
Standard Form of a Univariate Polynomial
A univariate polynomial is written in standard form when its terms are ordered from the term with the highest exponent of the variable down to the term with the lowest (or constant term, which has an exponent of 0). This form makes it easy to identify key features like the leading coefficient and degree. For example, $3x^2 + 5x + 1$ is in standard form, while $5x + 3x^2 + 1$ is not.
2
Leading Coefficient of a Polynomial
The leading coefficient is the numerical coefficient of the term with the highest exponent in a polynomial written in standard form. It can be positive, negative, zero (but if the leading term's coefficient is zero, that term does not exist, so we move to the next highest degree term), or a fraction. For the polynomial $-9y^3 + 23y^2 - 9y + 2$, the leading coefficient is -9.
3
Degree of a Univariate Polynomial
The degree of a univariate polynomial is the highest value of the exponent attached to the variable in any term of the polynomial. Constant terms (terms without a variable) have a degree of 0, and non-zero constant polynomials have a degree of 0, while the zero polynomial has an undefined or sometimes stated as -infinity degree. For $-9y^3 + 23y^2 - 9y + 2$, the highest exponent is 3, so the degree is 3.
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