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Chapter 5: Problem 11

Add $$ \left(7 t^{2}-3 t+6\right)+\left(2 t^{2}+8 t-9\right) $$

Short Answer

Expert verified
\(9t^2 + 5t - 3\)

Step by step solution

01

Identify Like Terms

Look at each polynomial and identify the like terms. Like terms are terms that have the same variable raised to the same power. In this case, the like terms are: \(7t^2\) and \(2t^2\), \(-3t\) and \(8t\), and \(6\) and \(-9\).
02

Combine Like Terms

Add the coefficients of the like terms together. For the \(t^2\) terms: \(7 + 2 = 9\). For the \(t\) terms: \(-3 + 8 = 5\). For the constant terms: \(6 - 9 = -3\).
03

Write the Simplified Expression

Combine the results from Step 2 to form the final polynomial. The terms are \(9t^2\), \(5t\), and \(-3\). Therefore, the simplified expression is \(9t^2 + 5t - 3\).

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Key Concepts

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

Like Terms
In polynomials, understanding 'like terms' is essential. Like terms are terms in an expression that have the same variable raised to the same power. For example, in the expression \((7t^2 - 3t + 6) + (2t^2 + 8t - 9)\), we need to identify which terms can be combined.
Here, \(7t^2\) and \(2t^2\) are like terms because they both have the variable \(t\) raised to the same power of 2. Similarly, \(-3t\) and \(+8t \) are like terms because they both have the variable \(t\) raised to the first power. Lastly, constants \(6\) and \(-9\) are also like terms.
Identifying like terms enables us to simplify expressions by combining them, which leads to a more compact form of the polynomial.
Coefficients
Coefficients are the numerical parts of the terms in a polynomial. When combining like terms, we add or subtract their coefficients. In our example, the coefficients are
  • 7 and 2 for the \(t^2\) terms,
  • -3 and 8 for the \(t\) terms,
  • and 6 and -9 for the constant terms.

Adding these coefficients results in:
  • For the \(t^2\) terms: \((7 + 2) = 9\),
  • For the \(-3t\) and \(8t \) terms: \((-3 + 8) = 5 \),
    • EFor the constant terms: \( (6 - 9) = -3 \).

The new coefficients give us the simplified polynomial: \(9t^2 + 5t - 3 \). Understanding coefficients is crucial as they determine how much of each term is present in the expression.
Simplified Expression
The final goal in polynomial addition is to write the expression in its simplified form. After combining like terms and their coefficients, we gather all the terms together into one polynomial.
In our example, we combined the coefficients and identified the like terms, which gave us: \((7t^2 + 2t^2) = 9t^2, \ ( -3t + 8t) = 5t\), and \( (6 - 9) = -3 \).
Thus, the simplified expression is \(9t^2 + 5t - 3\). A simplified expression is easier to work with and provides a clear, concise representation of the polynomial. Always aim to combine and reduce polynomials to their simplest form for clarity and efficiency in further calculations.

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