To determine the degree and coefficients of the polynomial [tex]\( y^3 - 5x^2 + 2x + 8 \)[/tex], we need to carefully analyze the polynomial step by step.
Step 1: Identify the degree of the polynomial.
The degree of a polynomial is the highest power of the variable (in this case, `x` or `y`) that appears in the expression with a non-zero coefficient.
In the polynomial [tex]\( y^3 - 5x^2 + 2x + 8 \)[/tex]:
- The term [tex]\( y^3 \)[/tex] has the variable [tex]\( y \)[/tex] raised to the power of 3.
- The term [tex]\( -5x^2 \)[/tex] has the variable [tex]\( x \)[/tex] raised to the power of 2.
- The term [tex]\( 2x \)[/tex] has the variable [tex]\( x \)[/tex] raised to the power of 1.
- The constant term [tex]\( 8 \)[/tex] does not contain any variable, which can be considered as the variable raised to the power of 0.
Here, the term [tex]\( y^3 \)[/tex] indicates that the polynomial has a degree of 3, since 3 is the highest power among the terms.
Step 2: Identify the coefficients of the polynomial.
The coefficients of a polynomial are the numerical factors of each term. For each term, we look at the number multiplying the variable.
For [tex]\( y^3 - 5x^2 + 2x + 8 \)[/tex]:
- The coefficient of [tex]\( y^3 \)[/tex] is 1 (since it can be written as [tex]\( 1 \cdot y^3 \)[/tex]).
- The coefficient of [tex]\( x^2 \)[/tex] is -5 (since it is [tex]\( -5 \cdot x^2 \)[/tex]).
- The coefficient of [tex]\( x \)[/tex] is 2 (since it is [tex]\( 2 \cdot x \)[/tex]).
- The constant term is 8, which can be viewed as [tex]\( 8 \cdot x^0 \)[/tex].
Therefore, the coefficients (considering polynomials in terms of [tex]\( x \)[/tex]) are [tex]\([1, -5, 2, 8]\)[/tex].
Conclusion:
The degree of the polynomial [tex]\( y^3 - 5x^2 + 2x + 8 \)[/tex] is 3, and the coefficients are [tex]\([1, -5, 2, 8]\)[/tex].
Thus, the final answer is:
[tex]\[
(3, [1, -5, 2, 8])
\][/tex]
