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Chapter 5: Problem 18

Add $$ \left(4 x^{5}-6 x^{3}-9 x+1\right)+\left(6 x^{3}+9 x^{2}+9 x\right) $$

Short Answer

Expert verified
The simplified expression is \(4x^5 + 9x^2 + 1\).

Step by step solution

01

- Align the Polynomials

Write down the polynomials in a vertical format, aligning the like terms (terms with the same power of x). \[ \begin{array}{r} 4x^5 - 6x^3 - 9x + 1 \ + 6x^3 + 9x^2 + 9x \end{array} \]
02

- Combine Like Terms

Combine the coefficients of like terms to simplify the expression: - The term with \(x^5\) is 4 (no like term to combine with) - The term with \(x^3\) is \(-6 + 6 = 0\) - The term with \(x^2\) is 9 (no like term to combine with) - The term with \(x\) is \(-9 + 9 = 0\) - The constant term is 1 (no like term to combine with)
03

- Write the Simplified Expression

Write down the simplified expression combining all like terms found: \[ 4x^5 + 9x^2 + 1 \]

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Key Concepts

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

Combining Like Terms
When adding polynomials, the first thing to do is to combine like terms. Like terms are terms that have the same variable raised to the same power. For example, in the polynomials given, the terms with \(x^3\) are like terms, because they both involve \(x\) raised to the 3rd power.
  • Identifying like terms simplifies the process of addition.
  • It is crucial to combine the coefficients of these like terms.
After identifying like terms, you can add or subtract their coefficients while keeping the variable part unchanged.
For example, if you have \(-6x^3\) and \(6x^3\), combining them means adding the coefficients (-6 and 6) together, resulting in 0. This tune helps in achieving a simpler polynomial expression.
Polynomial Addition
Polynomial addition involves summing multiple polynomials to get a new polynomial. To add polynomials correctly:
  • First, write them in a vertical format aligning all like terms vertically.
  • Next, combine the coefficients of the same power of \(x\).
For instance, in our example:
the coefficients are combined as follows:
  • \(4x^5\) remains as it is, since there鈥檚 no matching term.
  • \( -6x^3 + 6x^3 = 0\)
  • \(9x^2\) stands alone with no matching term.
  • \(-9x + 9x = 0\)
  • The constant 1 remains as it is.
This process helps in achieving a simplified polynomial after the addition.
Algebraic Expressions
Algebraic expressions are combinations of variables and constants using operations like addition, subtraction, multiplication, and division. Polynomials are a special type of algebraic expression consisting of terms where the exponents of the variables are non-negative integers.
  • A term in a polynomial can have coefficients, which are numerical factors multiplying the terms.
  • Being familiar with terms and their components makes it easier to perform operations like addition.
For example, in the expression \(4x^5 - 6x^3 - 9x + 1\):
  • \(4x^5\) is a term with a coefficient 4 and variable part \(x^5\).
  • Similarly, \(-6x^3\) has -6 as the coefficient and \(x^3\) as the variable part.
Understanding this structure makes combining like terms and adding polynomials straightforward.

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