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Chapter 5: Problem 47

Subtract. $$ \left(3+5 a+3 a^{2}-a^{3}\right)-\left(2+3 a-4 a^{2}+2 a^{3}\right) $$

Short Answer

Expert verified
1 + 2a + 7a^2 - 3a^3

Step by step solution

01

- Rewrite the Expression

Start by rewriting the given expression without parentheses: ewline (3 + 5a + 3a^2 - a^3) - (2 + 3a - 4a^2 + 2a^3)
02

- Distribute the Negative Sign

Distribute the negative sign across the second polynomial to combine like terms: ewline 3 + 5a + 3a^2 - a^3 - 2 - 3a + 4a^2 - 2a^3
03

- Combine Like Terms

Combine the like terms by grouping constant terms, terms with 'a', 'a^2', and 'a^3':ewline (3 - 2) + (5a - 3a) + (3a^2 + 4a^2) + (-a^3 - 2a^3)ewline This simplifies to: ewline 1 + 2a + 7a^2 - 3a^3

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combining Like Terms
In polynomials, 'like terms' are terms that have the same variable raised to the same power. When we subtract polynomials, we need to combine these like terms to simplify the expression. For instance, if you have the terms 5a and 3a, they are like terms because they both contain the variable 'a' raised to the same power (which is 1). Similarly, 3a^2 and 4a^2 are like terms, while terms such as 3 and 5a are not like terms since one is a constant, and the other includes a variable.
Distributing Negative Sign
When subtracting polynomials, it's crucial to distribute the negative sign (-) to all terms in the second polynomial before combining like terms. For the expression \(\firm{(3 + 5a + 3a^2 - a^3) - (2 + 3a - 4a^2 + 2a^3)}\), this means changing the signs of all terms inside the parentheses after the negative sign. Hence, it becomes:
  • \(-2\ becomes -2\)
  • \(+3a\ becomes -3a\)
  • \(-4a^2\ becomes +4a^2\)
  • \(+2a^3\ becomes -2a^3\)
This ensures accurate subtraction and prevents errors.
Polynomial Subtraction
Subtracting polynomials involves two main steps: distributing the negative sign and combining like terms. Using our example:
\(\firm{(3 + 5a + 3a^2 - a^3) - (2 + 3a - 4a^2 + 2a^3)}\), we first distribute the negative sign to get:
  • \(3 + 5a + 3a^2 - a^3 - 2 - 3a + 4a^2 - 2a^3\)
Next, combine the like terms:
  • Constants: \(3 - 2 = 1\)
  • 'a' terms: \(5a - 3a = 2a\)
  • 'a^2' terms: \(3a^2 + 4a^2 = 7a^2\)
  • 'a^3' terms: \(-a^3 - 2a^3 = -3a^3\)
Put it all together to get: \(1 + 2a + 7a^2 - 3a^3 \). This is the simplified form of the polynomial after subtraction.

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