Add or subtract, as indicated. −(8x3+x−3)+(2x3+x2)−(4x2+3x−$$1)-(8x^3+x-3$$) + $$(2x^3$+x^2$$) - $$(4x^2+3x-1)$$

Ch. R - Review of Basic Concepts

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Lial 13th Edition

Ch. R - Review of Basic Concepts

문제 28

1장, 문제 28

Add or subtract, as indicated. $- (8 x^{3} + x - 3) + (2 x^{3} + x^{2}) - (4 x^{2} + 3 x - 1)$

검증된 단계별 안내

First, rewrite the expression clearly, paying attention to the signs in front of each parenthesis: $-(8x^3 + x - 3) + (2x^3 + x^2) - (4x^2 + 3x - 1)$.

Next, distribute the negative signs to each term inside the parentheses where applicable: change the signs of all terms inside the first and third parentheses because of the leading minus signs.

After distribution, write out all the terms without parentheses: $-8x^3 - x + 3 + 2x^3 + x^2 - 4x^2 - 3x + 1$.

Now, group like terms together. Like terms are terms that have the same variable raised to the same power: group $x^3$ terms, $x^2$ terms, $x$ terms, and constant terms separately.

Finally, combine the coefficients of the like terms by adding or subtracting them accordingly to simplify the expression.

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Polynomial Addition and Subtraction

Adding or subtracting polynomials involves combining like terms, which are terms with the same variable raised to the same power. When subtracting, distribute the negative sign to each term inside the parentheses before combining. This ensures correct simplification of the expression.

Like Terms

Like terms have identical variable parts and exponents, such as 8x^3 and 2x^3. Only like terms can be added or subtracted directly by combining their coefficients. Recognizing like terms is essential for simplifying polynomial expressions accurately.

Distributive Property

The distributive property allows you to multiply a single term outside the parentheses by each term inside. For subtraction, it means applying a negative sign to every term within the parentheses, changing their signs before combining. This step is crucial to avoid errors in polynomial operations.

Add or subtract, as indicated. −(8x3+x−3)+(2x3+x2)−(4x2+3x−1)-(8x^3+x-3) + (2x^3+x^2) - (4x^2+3x-1)