Question

Simplify and write the resulting polynomial in descending order of degree. $$3 b-9 b^{4}+4 b^{4}-7 b$$

Short answer

Question: Simplify and write the polynomial expression $$3b - 9b^4 + 4b^4 - 7b$$ in descending order of degree. Answer: $$-5b^4 - 4b$$

Step-by-step solution

Group like terms together.

We have the polynomial $$3 b-9 b^{4}+4 b^{4}-7 b$$ We can rewrite it by grouping the like terms together: $$(-9b^4 + 4b^4) + (3b -7b)$$

Simplify coefficients of like terms.

Now that we have grouped the like terms together, it's time to simplify their coefficients. $$(-9b^4 + 4b^4) = -5b^4$$ and $$(3b - 7b) = -4b$$

Write the simplified polynomial in descending order

Now that we have simplified the coefficients, we need to write the polynomial in descending order of degree, which means putting the highest degree first and lower degrees in decreasing order. Our simplified polynomial is: $$-5b^4 - 4b$$

Key concepts

Like Terms

Understanding 'like terms' is crucial to effectively simplify polynomials. Like terms have the same variable raised to the same power. For instance, in the expression \(3b - 9b^4 + 4b^4 - 7b\), the terms \(-9b^4\) and \(4b^4\) are like terms because they both contain the variable \(b^4\). Similarly, \(3b\) and \(-7b\) are like terms as both have \(b\) raised to the power of 1. When simplifying, only combine like terms. This involves adding or subtracting their coefficients, while keeping the variable part unchanged. This process helps in reducing the expression to a more concise form, making it easier to manage and understand.

Descending Order

Descending order of a polynomial means arranging the polynomial's terms from the highest power to the lowest. For example, in the polynomial \(-5b^4 - 4b\), \(-5b^4\) comes first because it has the highest degree, which is 4, followed by \(-4b\), which has a degree of 1.Placing terms in descending order assists in understanding the behavior of polynomials, particularly when graphing or finding roots. It provides a consistent format that's easily recognizable and simple to work with. Remember, when writing your final answer, always check that the polynomial is in descending order, starting from the highest power to the lowest.

Polynomial Degree

The degree of a polynomial is the power of the highest term in the expression when the polynomial is written in standard form. For the polynomial \(-5b^4 - 4b\), the degree is 4 because the highest power of \(b\) present in the polynomial is 4.Identifying the degree is important because it provides valuable information about the polynomial's characteristics:

• The degree tells us the maximum number of solutions or roots the polynomial can have.
• The degree indicates the general shape of the polynomial's graph.
• Higher-degree polynomials can have more turning points compared to lower-degree ones.

Thus, understanding a polynomial's degree is fundamental in polynomial operations and analysis.