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

Q: What is the simplified form of the expression?

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.

Q: Is this expression a polynomial?

Yes. It is a multivariable polynomial in x and y because all exponents are nonnegative integers.

Q: What is the degree of the polynomial?

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

Q: How do I evaluate the expression for given x,y?

How do I evaluate the expression for given x,y?

Q: How to write the terms in standard order?

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

Q: How do I solve -2x^2+5y^3+2=0 for y?

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

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

What is the simplified form of the expression?

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

How do I identify like terms?

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

Is this expression a polynomial?

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

What is the degree of the polynomial?

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

Can the simplified expression be factored nicely?

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.

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

How to write the terms in standard order?

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

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

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

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