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Difficulty: MediumSimplifying Algebraic Expressions

Simplify the following algebraic expression completely. Write your final answer in standard form (highest to lowest power):

−2(n2−4n+3)+3n(n−2)-2(n^2 - 4n + 3) + 3n(n - 2)

Answer: n^2 + 2n - 6 / n^2+2n-6 / 1n^2 + 2n - 6 / n^2 + 2n -6 / n^2+2n - 6 / 1n^2+2n-6

Answer

n^2 + 2n - 6
The completely simplified form is determined by first distributing the multipliers into their respective parentheses. For the first group, -2(n^2 - 4n + 3) expands to -2n^2 + 8n - 6. For the second group, 3n(n - 2) expands to 3n^2 - 6n. Combining the expansions results in -2n^2 + 8n - 6 + 3n^2 - 6n. By adding the coefficients of like terms, the final standard form evaluates to n^2 + 2n - 6.

Step-by-Step Solution

1
Distribute -2 into the first polynomial, (n^2 - 4n + 3).
-2n^2 + 8n - 6
The distributive property requires multiplying the outside term by every term inside the parentheses. Care must be taken with the negative sign: -2 multiplied by -4n becomes positive 8n.
2
Distribute 3n into the second binomial, (n - 2).
3n^2 - 6n
Using the distributive property again, multiplying 3n by n gives 3n^2, and 3n multiplied by -2 gives -6n.
3
Group the like terms from both expansions together.
(-2n^2 + 3n^2) + (8n - 6n) - 6
Like terms are terms that share the exact same variables raised to the exact same power. Grouping them helps systematically combine the coefficients.
4
Simplify the grouped terms.
n^2 + 2n - 6
Adding the coefficients of the grouped terms: -2 + 3 = 1 for the n^2 terms, and 8 - 6 = 2 for the n terms.

Key Concept

Simplifying algebraic expressions via polynomial expansion and combining like terms.
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Topics

Simplifying Algebraic Expressions

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