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Chapter 5: Problem 38

Add. \(\left(6 x^{2}+5 x+7\right)\) and \(\left(x^{2}+6 x-3\right)\)

Short Answer

Expert verified
The sum is \(7x^2 + 11x + 4\).

Step by step solution

01

Identify the Like Terms

In order to add polynomials, we need to add the like terms. These are the terms that have the same variable part. In this case, the polynomials are \(6x^2 + 5x + 7\) and \(x^2 + 6x - 3\). The like terms we need to add are: \(6x^2\) and \(x^2\), \(5x\) and \(6x\), and \(7\) and \(-3\).
02

Add the Like Terms

Now add the like terms identified in Step 1: \[(6x^2 + x^2) + (5x + 6x) + (7 - 3)\]Calculate each pair:- For \(x^2\) terms: \(6x^2 + x^2 = 7x^2\)- For \(x\) terms: \(5x + 6x = 11x\)- For constant terms: \(7 - 3 = 4\)
03

Write the Resulting Polynomial

Combine all the sums from Step 2 to write the resulting polynomial. The final polynomial after adding is:\[7x^2 + 11x + 4\]

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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 essential in algebra, especially when adding polynomials. "Like terms" share the same variable and exponent. This similarity allows us to group and simplify expressions effectively. For example, in the expression
  • \(6x^2 + 5x + 7\),
  • the terms \(6x^2\) and \(x^2\) are like terms because they both involve \(x\) raised to the second power.
This applies to other terms as well.
  • The terms \(5x\) and \(6x\) are like terms since they both involve \(x\) to the first power,
  • and \(7\) and \(-3\) are also like terms as they don't have any variables attached, thus are constants.
In the process of adding polynomials, recognizing and grouping like terms ensures accuracy in calculations.
Polynomial Addition
Polynomial addition might sound complicated, but it involves straightforward steps. First, identify the like terms from each polynomial. Then, add the coefficients of these like terms together.
  • For the problem at hand, we identify like terms such as \(6x^2\) with \(x^2\), and \(5x\) with \(6x\).
  • Finally, include the constant terms \(7\) and \(-3\).
After identifying the terms, add their coefficients:
  • \(6x^2 + x^2 = 7x^2\),
  • \(5x + 6x = 11x\),
  • and \(7 - 3 = 4\).
This way, keep the variable part unchanged and only sum up the coefficients. The final result, \(7x^2 + 11x + 4\), represents the combined polynomial.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations. They can be simple or complex. In algebra, understanding how to manipulate and simplify them is crucial. Polynomials, a type of algebraic expression, consist of terms arranged in decreasing powers of a variable.

For instance, our given polynomials \((6x^2 + 5x + 7)\) and \((x^2 + 6x - 3)\) are algebraic expressions containing three terms each.
  • Terms involving a variable to a power, like \(x^2\) or \(x\), are known as variable terms.
  • Those without a variable, such as the constants \(7\) and \(-3\), are called constant terms.
Knowing how to interpret and work with these expressions is foundational in algebra. As students become more familiar with such expressions, operations like addition become considerably simpler.

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Most popular questions from this chapter

If \(P(x)\) is the polynomial given, find a. \(P(a),\) b. \(P(-x),\) and c. \(P(x+h)\). \(P(x)=8 x+3\)

Use substitution to factor each polynomial completely. See Examples 11 and 12. $$ x^{4}+x^{2}-6 $$

Factor each polynomial completely. See Examples 1 through 12. $$ 2 x^{6}+3 x^{3}-9 $$

Factor. Assume that variables used as exponents represent positive integers. $$ x^{2 n}-7 x^{n}+12 $$

Factor each trinomial. See Examples 5 through 10. $$ 3 x^{2}+14 x+15 $$
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