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Chapter 5: Problem 43

Add. $$ \begin{aligned} &6 a^{4}+9 a^{2}+a\\\ &+2 a^{4}-13 a^{2}+a \end{aligned} $$

Short Answer

Expert verified
The sum is \(8a^4 - 4a^2 + 2a\).

Step by step solution

01

Identify Like Terms

Write down the polynomial terms from both expressions that will be added. We have \(6a^4, 9a^2, a\) from the first expression, and \(2a^4, -13a^2, a\) from the second expression. Identify like terms: \(6a^4\) and \(2a^4\) are like terms, \(9a^2\) and \(-13a^2\) are like terms, and the 'a' terms are \(a\) and \(a\).
02

Add the Coefficients of Like Terms

Add the coefficients of the terms with the same power. For the \(a^4\) terms: \(6 + 2 = 8\). For the \(a^2\) terms: \(9 - 13 = -4\). For the \(a\) terms: \(1 + 1 = 2\).
03

Write the Simplified Polynomial

Using the results from Step 2, write the simplified expression. The sum of the polynomials is \(8a^4 - 4a^2 + 2a\).

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Key Concepts

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

Understanding Like Terms
In polynomials, determining which parts of the expression are like terms is the first step toward simplifying an expression. Like terms are terms that have the same variable raised to the same power. The coefficients can be different, but the variable-part must match exactly.
For instance, consider the expression from our exercise earlier, which includes terms like \(6a^4\), \(9a^2\), and \(a\). Each of these are distinct due to their respective exponents. However, when matching like terms from another polynomial, such as \(2a^4\), it鈥檚 clear that \(6a^4\) and \(2a^4\) are like terms because they both contain \(a^4\).
To determine like terms,
  • Focus on the variable and its exponent.
  • Terms with the same variable and exponent are like terms.
  • Combine like terms to simplify the expression.
Understanding Coefficients
Coefficients are the numerical part of terms that appear with variables in a polynomial. In simplifying expressions, adding or subtracting the coefficients of like terms is crucial.

In our exercise, for example, the coefficient of \(a^4\) is 6 in one term and 2 in another. When adding polynomials together, like terms align, and you perform addition or subtraction on their coefficients. This gives you a new coefficient of \(8\) for \(a^4\) since \(6 + 2 = 8\).
  • The coefficient is always the number preceeding the variables.
  • It tells you how many of the term there are.
  • When adding like terms, sum their coefficients. When subtracting, find the difference.
Simplifying Expressions
Simplifying expressions involves combining like terms and making an expression as concise as possible. Once you have identified and combined the coefficients of like terms, you can easily rewrite the expression.
The goal is to reduce the expression to its simplest form, where like terms have been appropriately combined and organized. For each group of like terms in our exercise
  • Identify the formula and combine the coefficients: as we did with \(8a^4\), \(-4a^2\), and \(2a\).
  • The simplified polynomial is \(8a^4 - 4a^2 + 2a\), showing the reduction from a longer form to the short outcome.
  • This final step removes unnecessary repetition and provides a clear, clean representation of the result.
Understanding these basics helps build a foundation for working efficiently with any polynomial expressions.

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Most popular questions from this chapter

A company purchased two vehicles for its sales force to use. The following functions give the respective values of the vehicles after \(x\) years. Toyota Camry LE: \(T(x)=-2,500 x+21,175\) Ford Explorer XLT: \(F(x)=-2,900 x+31,190\) A. Find one polynomial function \(V\) that will give the combined value of both cars after \(x\) years. B. Use your answer in part (a) to find the combined value of the two cars after 3 years.

Solve: \(\frac{a^{3}}{65}-\frac{a^{2}}{30}-\frac{a}{78}=0\)

Storage Tanks. The volume \(V(r)\) of the gasoline storage tank, in cubic feet, is given by the polynomial function \(V(r)=4.2 r^{3}+37.7 r^{2},\) where \(r\) is the radius in feet of the cylindrical part of the tank. What is the capacity of the tank if its radius is 4 feet? (PICTURE NOT COPY)

Construct a table of values for each polynomial function using the given values for \(x .\) Then graph the function and find its domain and range. $$ \begin{aligned} &f(x)=-x^{3}-x^{2}+6 x\\\ &x=-4,-3,-2,-1,0,1,2,3 \end{aligned} $$ $$ \begin{array}{|r|r|} \hline x & f(x) \\ \hline-4 & \\ -3 & \\ -2 & \\ -1 & \\ 0 & \\ 1 & \\ 2 & 3 & \\ \hline \end{array} $$

Explain why \(f(x)=\frac{1}{x+1}\) is not a polynomial function.
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