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Chapter 5: Problem 68

Add. $$ \begin{array}{c}{-6 y^{3}+8 y+5} \\ {9 y^{3}+4 y-6} \\ \hline\end{array} $$

Short Answer

Expert verified
The result is \(3y^3 + 12y - 1\).

Step by step solution

01

Arrange the Terms

Align the terms of the two polynomials so that like terms are in the same column: \[\begin{array}{c}{-6y^{3} + 8y + 5} \ {+ \, 9y^{3} + 4y - 6} \ \hline\end{array}\]
02

Add the Cubic Terms

Add the coefficients of the cubic terms (\(y^3\) terms):\[-6y^3 + 9y^3 = 3y^3\]
03

Add the Linear Terms

Add the coefficients of the linear terms (\(y\) terms):\[8y + 4y = 12y\]
04

Add the Constant Terms

Add the constant terms:\[5 - 6 = -1\]
05

Combine the Results

Combine the sums of each term to get the result:\[3y^3 + 12y - 1\]

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Key Concepts

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

Understanding Cubic Terms in Polynomials
Cubic terms are the parts of a polynomial where the variable is raised to the power of three. In this exercise, the cubic terms are \(-6y^3\) and \(9y^3\). When adding cubic terms, we focus on their coefficients. \(-6 + 9 = 3\). Thus, the sum of the cubic terms in our example is \(3y^3\). Remember, cubic terms help shape the curve of the polynomial graph, making them crucial in understanding the function's behavior.
Comprehending Linear Terms in Polynomial Addition
Linear terms are where the variable is not raised to any power higher than one. In our exercise, the linear terms are \(8y\) and \(4y\). To combine these, we add their coefficients: \(8 + 4 = 12\). This gives us the term \(12y\). Linear terms indicate the slope of a line in the polynomial. The higher the coefficient, the steeper the line.
Identifying and Adding Constant Terms
Constant terms are numbers without any variables. They are also known as the zero power terms. In the given polynomials, the constants are 5 and -6. When adding constants, simply perform basic arithmetic: \(5 - 6 = -1\). The result, \(-1\), is our constant term in the new polynomial. Constant terms shift the polynomial graph up or down.

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