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Chapter 5: Problem 31

Simplify each polynomial. See Example 5 . $$ -7 y^{2}-11 y^{3}-4+y^{3}-y^{2} $$

Short Answer

Expert verified
The simplified polynomial is \(-10y^3 - 8y^2 - 4\).

Step by step solution

01

Identify Like Terms

First, identify the terms with the same degree in the polynomial \(-7y^2 - 11y^3 - 4 + y^3 - y^2\). The like terms are \(-11y^3\) and \(y^3\); \(-7y^2\) and \(-y^2\); and the constant term is \(-4\).
02

Combine Like Terms

Combine the like terms. First, simplify the cubic terms: \(-11y^3 + y^3 = -10y^3\). Next, simplify the quadratic terms: \(-7y^2 - y^2 = -8y^2\). Finally, the constant term remains as is, i.e., \(-4\).
03

Write the Simplified Polynomial

Write down the simplified polynomial by collecting all the simplified terms: \(-10y^3 - 8y^2 - 4\). This is the polynomial in its simplest form.

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Key Concepts

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

Like Terms
In algebra, terms are considered "like terms" when they have exactly the same variable(s) raised to the same power. This means that the coefficients can be different, but as long as the variable and its exponent match, they can be combined. Identifying like terms is crucial because it allows you to simplify a polynomial by combining those terms, effectively reducing the number of terms you need to work with.

When looking at the polynomial \(-7y^2 - 11y^3 - 4 + y^3 - y^2\), we can safely say:
  • \(-11y^3\) and \(+y^3\) are like terms because they both involve \(y\) raised to the power of 3.
  • \(-7y^2\) and \(-y^2\) are also like terms, involving \(y\) to the power of 2.
  • \(-4\) is a constant term and stands on its own.
Learning to identify like terms is the first step towards simplifying polynomials, as it sets the stage for combining these terms in the next steps.
Combining Like Terms
Once you've identified like terms, the next step is to combine them. Combining like terms simply means adding or subtracting the coefficients while keeping the variable part the same.

For example, in the polynomial \(-7y^2 - 11y^3 - 4 + y^3 - y^2\):
  • The terms \(-11y^3\) and \(+y^3\) can be combined to get \(-10y^3\). This is done by adding their coefficients: \(-11 + 1 = -10\).
  • Similarly, the terms \(-7y^2\) and \(-y^2\) combine to give \(-8y^2\). Again, add their coefficients: \(-7 - 1 = -8\).
By performing these operations, you bring the polynomial closer to its simplest form without altering its value. Note that the constant term \(-4\) remains as it is because it doesn't have any like terms to combine with.
Simplified Polynomial
A simplified polynomial is the result of combining all possible like terms into their simplest form. It allows easier problem-solving and better understanding of the polynomial's behavior.

Let's see the outcome of simplifying the polynomial \(-7y^2 - 11y^3 - 4 + y^3 - y^2\). After combining like terms, we have:
  • \(-10y^3\) from the cubic terms.
  • \(-8y^2\) from the quadratic terms.
  • \(-4\), the constant term, stays the same.
This gives us the simplified polynomial: \(-10y^3 - 8y^2 - 4\). This form has fewer terms and is much easier to handle in further calculations or when solving equations. Always remember, simplification of a polynomial doesn't change its root values or its fundamental properties; it just streamlines the expression.

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Most popular questions from this chapter

Look Alikes... A. \(\quad 9 x^{5}-7 x^{3}+x-1\) \(+6 x^{5}-7 x^{3}+3 x+1\) \begin{tabular}{l} B. \(9 x^{5}-7 x^{3}+x-1\) \\ \(-\left(6 x^{5}-7 x^{3}+3 x-1\right)\) \\ \hline \end{tabular}

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