For the polynomial 8 - y + (1/2)y^3 - 3y^2, write: (a) constant term (b) coefficient of y^2 (c) the degree of the polynomial.
Understand the Problem
The question is asking for specific characteristics of the polynomial provided. It requests the identification of the constant term, the coefficient of the y² term, and the degree of the polynomial.
Answer
(a) $8$; (b) $-3$; (c) $3$
Answer for screen readers
(a) The constant term is $8$.
(b) The coefficient of $y^2$ is $-3$.
(c) The degree of the polynomial is $3$.
Steps to Solve
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Identify the constant term The constant term in a polynomial is the term that does not contain any variable. In the polynomial $8 - y + \frac{1}{2} y^3 - 3y^2$, the constant term is $8$.
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Determine the coefficient of the (y^2) term The coefficient of the term involving $y^2$ is the number multiplying the $y^2$. In the polynomial, the term is $-3y^2$, so the coefficient of $y^2$ is $-3$.
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Find the degree of the polynomial The degree of a polynomial is the highest power of the variable present. The polynomial contains the term $\frac{1}{2}y^3$, which has a degree of $3$. Therefore, the degree of the polynomial is $3$.
(a) The constant term is $8$.
(b) The coefficient of $y^2$ is $-3$.
(c) The degree of the polynomial is $3$.
More Information
Polynomials are expressions consisting of variables raised to non-negative integer powers and combined using addition, subtraction, and multiplication. The constant term is crucial for determining the value of the polynomial when all variable components are set to zero. The degree indicates the polynomial's highest level of complexity in terms of behavior and shape on a graph.
Tips
- Confusing the constant term with the coefficient of a variable term. Always look for the term without variables for the constant.
- Misidentifying the highest power of a variable; be careful to check all terms for the highest exponent.
