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Chapter 5: Problem 21

Add the polynomials. $$\left(7 x^{2}-11 x\right)+\left(3 x^{2}-x\right)$$

Short Answer

Expert verified
The result of adding the two polynomials \( \left(7 x^{2}-11 x\right) \) and \( \left(3 x^{2}-x\right) \) is \( 10x^{2}-12x \).

Step by step solution

01

Identify Like Terms

Like terms are terms that have the same variable and exponent. Looking at the two polynomials, it is seen that \(7 x^{2}\) and \(3 x^{2}\) are like terms because they both contain \(x^{2}\). Similarly, \(-11 x\) and \(-x\) are like terms because they both contain \(x\) with the power of 1.
02

Add the Coefficients of a

Add the coefficients of the \(x^{2}\) terms together. It's important to remember that the term 'coefficient' refers to the numerical part of the term. The coefficients in the \(x^{2}\) terms are 7 and 3, add these together to get \(7 + 3 = 10\). So the new \(x^{2}\) term is \(10x^{2}\).
03

Add the Coefficients of b

Now, add together the coefficients of the \(x\) terms. The coefficients in the \(x\) terms are -11 and -1 i.e., \(-11 - 1 = -12\). So, the new \(x\) term is \(-12x\).
04

Combine the New Terms

Finally, combine the new \(x^{2}\) term and the new \(x\) term to get the result of the addition of the two polynomials. Thus, \((7 x^{2}-11 x)+(3 x^{2}-x) = 10x^{2}-12x\)

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Key Concepts

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

Like Terms
In the context of polynomials, understanding "like terms" is essential. A like term is one that shares the same variable raised to the same power. This is crucial when performing operations such as addition and subtraction with polynomials.
In the problem given, we identify the like terms by examining the powers of the variables:
  • The terms \(7x^{2}\) and \(3x^{2}\) are like terms because they both have the variable \(x\) to the power of 2.
  • Similarly, \(-11x\) and \(-x\) are also like terms, both having the variable \(x\), even though \(-x\) is known as \(-1x\).
Recognizing like terms allows you to simplify expressions by combining these terms, leading us to the next step: adding their coefficients.
Coefficients
Once like terms in a polynomial are identified, the next step involves working with the coefficients. The coefficient in a term is the numerical factor that multiplies the variable. It is crucial to combine like terms by adding or subtracting their coefficients.
For example, consider the terms \(7x^{2}\) and \(3x^{2}\). These have coefficients of 7 and 3, respectively. To combine them, you simply add the coefficients: \(7 + 3 = 10\). This results in \(10x^{2}\).
  • Similarly, for \(-11x\) and \(-x\): their coefficients are \(-11\) and \(-1\) (because \(-x\) is the same as \(-1x\)).
  • Adding these gives \(-11 - 1 = -12\), which means the combined term is \(-12x\).
Understanding coefficients is fundamental not just for adding like terms, but for manipulating polynomials in various operations, allowing you to consolidate terms accurately.
Polynomial Simplification
Polynomial simplification is the process of reducing a polynomial to its simplest form. After identifying like terms and working with their coefficients, we can simplify the expression further by combining terms.
In our exercise, we started with two separate polynomial expressions: \( (7x^{2} - 11x) \) and \( (3x^{2} - x) \). After adding like terms:
  • We combined \(7x^{2}\) and \(3x^{2}\) to get \(10x^{2}\).
  • We combined \(-11x\) and \(-x\) to get \(-12x\).
The result is a single, simplified polynomial: \(10x^{2} - 12x\). This simplification process is crucial as it makes expressions easier to understand and operate on further, especially when solving equations or integrating polynomials.

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Most popular questions from this chapter

In each exercise, find the product. $$(x+3)\left(x^{2}+5\right)$$

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