Q. \(x^{2} + 2y^{3} – 3x^{2} + 1 + 5 – 4 + 3y^{3}\)

Answer

Rewrite and combine like terms:
\[
x^2+2y^3-3x^2+1+5-4+3y^3
=(x^2-3x^2)+(2y^3+3y^3)+(1+5-4)
=-2x^2+5y^3+2.
\]

Final result: \(\boxed{-2x^2+5y^3+2}\).

Detailed Explanation

Write the expression to simplify:

\[
x^2 + 2y^3 – 3x^2 + 1 + 5 – 4 + 3y^3
\]
Group like terms (collect powers of \(x\), powers of \(y\), and constants):

\[
(x^2 – 3x^2) + (2y^3 + 3y^3) + (1 + 5 – 4)
\]
Simplify each group separately:

\[
x^2 – 3x^2 = -2x^2
\]

\[
2y^3 + 3y^3 = 5y^3
\]

\[
1 + 5 – 4 = 2
\]
Combine the simplified groups to get the final simplified expression:

\[
-2x^2 + 5y^3 + 2
\]

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FAQs

Q1: What is the simplified form of the expression?

A1: Combine like terms: \(x^2-3x^2=-2x^2\), \(2y^3+3y^3=5y^3\), \(1+5-4=2\). Simplified: \(-2x^2+5y^3+2\).

Q2: How do I identify like terms?

A2: Like terms have the same variable(s) with identical exponents. Here \(x^2\) terms combine, \(y^3\) terms combine, and the constant terms combine.

Q3: Is this expression a polynomial?

A3: Yes. It is a multivariable polynomial in \(x\) and \(y\) because all exponents are nonnegative integers.

Q4: What is the degree of the polynomial?

A4: The highest total degree is 3 (from the \(y^3\) term), so the polynomial’s degree is 3.

Q5: Can the simplified expression be factored nicely?

A5: There is no nontrivial common factor across all terms. You can rewrite as \(5y^3-2x^2+2\) or factor out \(-1\), but no simple integer-factorization exists.

Q6: How do I evaluate the expression for given \(x,y\)?

A6: Substitute into \(-2x^2+5y^3+2\) and compute. Example: \(x=1,y=0\) gives \(-2(1)^2+5(0)^3+2=0\).

Q7: How to write the terms in standard order?

A7: Often order by descending degree: \(5y^3-2x^2+2\) (degree 3 term first, then degree 2, then constant).

Q8: How do I solve \(-2x^2+5y^3+2=0\) for \(y\)?

A8: Isolate \(y^3\): \(y^3=\dfrac{2x^2-2}{5}\). Then \(y=\left(\dfrac{2x^2-2}{5}\right)^{1/3}\) (real cube root).

Q9: Could the original formatting be misread?

A9: Yes - check for missing operators or parentheses. Confirm intended exponents: likely \(x^2+2y^3-3x^2+1+5-4+3y^3\).

Q10: Is factoring by grouping applicable?

A10: Not effectively here; grouping pairs won’t produce common binomial factors because terms involve different variables and degrees.

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