Simplify polynomial expressions

Simplification involves rewriting a polynomial as an equivalent expression by applying the distributive property and combining like terms—terms with the same variable factors and exponents—while preserving coefficients, signs, and constants. The resulting expression is typically written in standard form, with like terms combined and unlike terms kept separate; this understanding supports evaluation, solving equations, factoring, and interpreting algebraic relationships, without extending to advanced polynomial identities or operations.

Detailed Explanation: Simplify polynomial expressions

To simplify a polynomial expression:

Use the distributive property to remove parentheses.
Identify like terms—terms with the same variable and exponent.
Combine the coefficients of like terms.
Write the result in standard form, from the greatest exponent to the smallest.

Example

Simplify:

3(2x^2-x+4)+2x^2+5x-1

Step 1: Distribute the $3$ to every term inside the parentheses:

3(2x^2)-3(x)+3(4)+2x^2+5x-1

6x^2-3x+12+2x^2+5x-1

Step 2: Group like terms:

(6x^2+2x^2)+(-3x+5x)+(12-1)

Step 3: Combine each group:

8x^2+2x+11

Therefore, the simplified expression is

\boxed{8x^2+2x+11}

Remember: $x^2$ terms combine with $x^2$ terms, $x$ terms combine with $x$ terms, and constants combine with constants.

Learn by doing: Simplify polynomial expressions

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Like Terms - Combine - Order N - Whole - One Variable - Expression to Pair

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