Question

Simplify and write the resulting polynomial in descending order of degree. $$4 m^{7}-9 m^{5}-12 m^{5}+7 m^{7}$$

Short answer

Question: Simplify the polynomial $$4 m^{7}-9 m^{5}-12 m^{5}+7 m^{7}$$ and rewrite it in descending order of degree. Answer: The simplified polynomial is $$11 m^{7} - 21 m^{5}$$.

Step-by-step solution

Identify like terms

Identify all like terms in the given polynomial: $$4 m^{7}-9 m^{5}-12 m^{5}+7 m^{7}$$ Like terms are terms with the same variable and exponent, so in this case, the like terms are the terms with the same degree: \(4m^7\) and \(7m^7\) are like terms, and \(-9m^5\) and \(-12m^5\) are like terms.

Combine like terms

Now, let's combine the like terms by adding or subtracting them: $$(4 m^{7} + 7 m^{7}) + (-9 m^{5} -12 m^{5})$$ $$(4+7) m^{7} + (-9-12) m^{5}$$ $$11 m^{7} - 21 m^{5}$$

Write the polynomial in descending order of degree

The resulting polynomial is already in descending order of degree (since the term with the highest degree, \(11m^7\), comes first, followed by the term with the next highest degree, \(-21m^5\)). So the final simplified polynomial is: $$11 m^{7} - 21 m^{5}$$

Key concepts

like terms

Understanding what like terms are is the first step to simplifying a polynomial. Like terms are terms in an expression that have the exact same variable raised to the same power. This means that the coefficients can be different, but the variables and their exponents must match exactly. For example, in the polynomial \(4m^7 - 9m^5 - 12m^5 + 7m^7\), the terms \(4m^7\) and \(7m^7\) are like terms because they both involve \(m\) raised to the power of 7. Similarly, \(-9m^5\) and \(-12m^5\) are like terms because they both involve \(m\) to the 5th power. Identifying these like terms correctly is crucial as it allows us to combine them in the next steps.
When you look for like terms:

• Check that they have the same variables.
• Ensure those variables have the same exponents.
• Only the coefficients (numbers in front of the variables) can differ.

Recognizing like terms simplifies the process of combining them in the next step.

combining like terms

Once like terms are identified, the next step is to combine them. This process involves adding or subtracting the coefficients of the like terms. The variable and its exponent remain unchanged; only the numerical coefficients are adjusted. Take the example from our polynomial \(4m^7 - 9m^5 - 12m^5 + 7m^7\). By combining like terms, we perform the following:

• For \(4m^7\) and \(7m^7\), combine to get \((4+7)m^7 = 11m^7\).
• For \(-9m^5\) and \(-12m^5\), combine to get \((-9-12)m^5 = -21m^5\).

The result is \(11m^7 - 21m^5\).
This step requires careful addition or subtraction of coefficients while maintaining the identical variable part. This is important because it helps in reducing complexity and preparing the polynomial for arranging it in the descending order of degrees.

descending order of degree

After combining the like terms, the final step is to arrange the polynomial in the descending order of degree. This essentially means writing the terms from the highest power to the lowest. The degree of a term in a polynomial is determined by the exponent of the variable. Therefore, organizing the terms involves focusing on these exponents.
In the example polynomial \(11m^7 - 21m^5\), the term \(11m^7\) comes first because 7 is the highest exponent in the polynomial. The term \(-21m^5\) follows because 5 is the next highest exponent.

• List each term starting with the highest exponent.
• Proceed to the next higher term until all terms are listed from highest to lowest degree.
• Remember, this ordering makes it easier to read and is a conventional approach in mathematics.

Writing polynomials in this order helps in presenting them in a standard form, making further operations, such as addition or subtraction with other polynomials, straightforward.