Problem 44 - Beginning and Intermediate Algebra

Question

Add \(9 a^{4}+5.3 a^{3}-7 a^{2}-1.2 a+6\) \(+-8 a^{4}-2.8 a^{3}+4 a^{2}-3.9 a+5\)

Step-by-Step Solution

Answer

The short answer is: \(a^4 + 2.5a^3 - 3a^2 - 5.1a + 11\)

Step 1: Identify the like terms

In order to add the polynomials, we need to identify the like terms in both polynomials. Like terms are the terms with the same variables and exponents. In our case, the like terms in these two polynomials are: \[9a^4\] and \[-8a^4\] \[5.3a^3\] and \[-2.8a^3\] \[-7a^2\] and \[4a^2\] \[-1.2a\] and \[-3.9a\] \[6\] and \[5\]

Step 2: Combine the like terms

Next, we need to combine the like terms. To do this, simply add the coefficients of the like terms: \(9a^4 - 8a^4 = a^4\) \(5.3a^3 - 2.8a^3 = 2.5a^3\) \(-7a^2 + 4a^2 = -3a^2\) \(-1.2a - 3.9a = -5.1a\) \(6 + 5 = 11\)

Step 3: Write the simplified expression

Now, we can write down the simplified expression by combining the resulting terms from the previous step: \[a^4 + 2.5a^3 - 3a^2 - 5.1a + 11\] So, the sum of the given polynomials is: \[ a^4 + 2.5a^3 - 3a^2 - 5.1a + 11 \]

Key Concepts

Like Terms in Polynomials

When adding polynomials, it's crucial to identify **like terms**. Like terms are those that have the same variable raised to the same power. For example, in the polynomial expression, terms like \(9a^4\) and \(-8a^4\) are like terms, because they both contain the variable \(a\) raised to the power of 4. Finding like terms is the first step in adding polynomials correctly.If you're dealing with several terms, you can:

• Group terms that have the same variable and exponent.
• Make sure you recognize all like terms; sometimes they can be hiding between many terms.

Understanding and identifying like terms helps in simplifying polynomial expressions easily.

Polynomial Addition Steps

Adding polynomials might seem complex at first, but it's a straightforward process once you break it down into steps. Here’s a step-by-step guide:First, identify the like terms as discussed previously. Once like terms are matched, the next steps involve:

• Adding the coefficients of each pair of like terms together.
• This doesn’t change the variable part; it only adjusts the numeric part.
• For example, if you have \(9a^4\) and \(-8a^4\), then you calculate \(9 - 8\) for the coefficients, resulting in \(a^4\).

These steps help ensure all amounts are accounted for correctly in your final expression. After performing these operations for each set of like terms in the polynomials, you end up with a clear and combined result.

Simplifying Polynomials

Simplifying polynomials means to reduce the expression to its simplest form while ensuring no further simplification is possible. When you've successfully added the like terms, by following the polynomial addition steps:- Combine all like terms to get the final polynomial.- Ensure there are no like terms left uncombined, which may indicate a mistake.For our example, we've combined all our terms to achieve: \[ a^4 + 2.5a^3 - 3a^2 - 5.1a + 11 \]No further simplification can happen here, as all terms are unique in terms of their exponents.Remember, a simplified polynomial is neater and easier to work with, making it easier to further evaluate or use in mathematical operations.