Perform the indicated operations. 2(3r2+4r+2)−3(−r2+4r−$$5)2(3r^2+4r+2$$) - $$3(-r^2+4r-5)$$

Ch. R - Review of Basic Concepts

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

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Lial 13th Edition

Ch. R - Review of Basic Concepts

Problem 66

Chapter 1, Problem 66

Perform the indicated operations. $2 (3 r^{2} + 4 r + 2) - 3 (- r^{2} + 4 r - 5)$

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Start by distributing the constants outside the parentheses to each term inside the parentheses. For the first part, multiply 2 by each term in the expression $3r^2 + 4r + 2$, and for the second part, multiply -3 by each term in the expression $-r^2 + 4r - 5$. This gives you:

\[ 2 \times 3r^2 + 2 \times 4r + 2 \times 2 - 3 \times (-r^2) - 3 \times 4r - 3 \times (-5) \]

Simplify each multiplication to rewrite the expression without parentheses:

\[ 6r^2 + 8r + 4 + 3r^2 - 12r + 15 \]

Combine like terms by adding or subtracting the coefficients of $r^2$ terms, $r$ terms, and constant terms separately to get the simplified expression.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Distributive Property

The distributive property allows you to multiply a single term by each term inside a parenthesis. For example, a(b + c) = ab + ac. This property is essential for expanding expressions like 2(3r^2 + 4r + 2) by multiplying 2 with each term inside the parentheses.

Combining Like Terms

Combining like terms involves adding or subtracting terms that have the same variable raised to the same power. For instance, 3r^2 and -3r^2 are like terms. After distributing, you combine these terms to simplify the expression into a single polynomial.

Handling Negative Signs

When subtracting an expression, the negative sign affects all terms inside the parentheses. For example, -3(-r^2 + 4r - 5) requires multiplying -3 by each term, changing signs accordingly. Properly managing these signs is crucial to avoid errors in simplification.

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