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Chapter 5: Problem 115

Is it easier to evaluate a polynomial before or after like terms have been combined? Why?

Short Answer

Expert verified
It is easier to evaluate a polynomial after combining like terms because it simplifies calculations.

Step by step solution

01

Understanding Like Terms

Like terms in a polynomial are terms that have the same variable raised to the same power. For example, in the polynomial expression \(3x^2 + 5x + 4x^2 + 2\), the terms \(3x^2\) and \(4x^2\) are like terms.
02

Combining Like Terms

To combine like terms, add or subtract the coefficients of the like terms. For the example \(3x^2 + 5x + 4x^2 + 2\), combine \(3x^2\) and \(4x^2\) to get \((3+4)x^2 = 7x^2\). The simplified polynomial is \(7x^2 + 5x + 2\).
03

Evaluating the Polynomial Before Combining

Evaluate the polynomial before combining like terms. For instance, if \(x = 1\), the original expression \(3x^2 + 5x + 4x^2 + 2\) becomes \(3(1)^2 + 5(1) + 4(1)^2 + 2 = 3 + 5 + 4 + 2 = 14\).
04

Evaluating the Polynomial After Combining

Evaluate the simplified polynomial. Using \(x = 1\) again, the simplified expression \(7x^2 + 5x + 2\) becomes \(7(1)^2 + 5(1) + 2 = 7 + 5 + 2 = 14\).
05

Conclusion

It is easier to evaluate a polynomial after combining like terms because it reduces the number of terms and simplifies calculations. The result remains the same whether the polynomial is evaluated before or after combining like terms.

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Key Concepts

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

Like Terms
Understanding like terms is crucial for simplifying polynomial expressions. Like terms in a polynomial have identical variables raised to the same power. For example, in the polynomial expression \(3x^2 + 5x + 4x^2 + 2\), the terms \(3x^2\) and \(4x^2\) are like terms because they both contain the variable \(x\) raised to the power of 2. Recognizing like terms allows us to combine them, making our polynomials simpler and more manageable to work with.
Combining Like Terms
Combining like terms involves adding or subtracting their coefficients. This step is essential because it simplifies the polynomial, reducing the number of terms we need to manage. For example, given the expression \(3x^2 + 5x + 4x^2 + 2\), we can combine \(3x^2\) and \(4x^2\):
  • First, add their coefficients: \((3+4)x^2 = 7x^2\).
  • Next, the simplified polynomial becomes \(7x^2 + 5x + 2\).
Combining like terms not only makes the polynomial easier to evaluate but also reduces the risk of making errors in subsequent steps.
Evaluating Expressions
Evaluating a polynomial means finding its value for a specific variable. It's easier to do this after combining like terms, as shown in our example with \(x = 1\). First, let's see how it works before combining like terms:
  • The original expression is \(3x^2 + 5x + 4x^2 + 2\).
  • When \(x = 1\), the expression becomes \(3(1)^2 + 5(1) + 4(1)^2 + 2\), which simplifies to \(3 + 5 + 4 + 2 = 14\).

Now, let's evaluate the polynomial after combining like terms:
  • The simplified expression is \(7x^2 + 5x + 2\).
  • Again, using \(x = 1\), it becomes \(7(1)^2 + 5(1) + 2 = 7 + 5 + 2 = 14\).

In both scenarios, the result is the same. However, it's much simpler to evaluate the polynomial after combining like terms because fewer steps are involved. This makes the entire process more efficient and less prone to mistakes.

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