Problem 7 \(\left(7 x^6+2 x^5-3 x^2+9 x\ri... [FREE SOLUTION]

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BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Ron Larson, Laurie Boswell

$Math Studyset 91影视 Explanations$ Math

1 Edition

Chapter 4: Problem 7

$\left(7 x^6+2 x^5-3 x^2+9 x\right)+\left(5 x^5+8 x^3-6 x^2+2 x-5\right)$

Short Answer

Expert verified

The simplified expression is $7 x^6 + 7 x^5 + 8 x^3 - 9 x^2 + 11 x - 5$.

Step by step solution

Identify Like Terms

Firstly, identify the 'like terms' in the two polynomials. 'Like terms' in this instance are: $7 x^6$ and $0 x^6$, $2 x^5$ and $5 x^5$, $-3 x^2$ and $-6 x^2$, $9 x$ and $2 x$, and finally, $0$ and $-5$ (as there is no constant term in the first polynomial, consider it as $0$. The terms with $x^3$ and $x^6$ in the second polynomial have no matching pair in the first polynomial, list them separately.

Add Like Terms

Next, add together each pair of 'like terms'. This gives: $7 x^6 + 0x^6 = 7 x^6$, $2 x^5 + 5x^5 = 7 x^5$, $-3 x^2 - 6x^2 = -9 x^2$, $9 x + 2x = 11 x$, and $0 - 5 = -5$. The exclusive terms of the second polynomial remain as they are: $0 x^3 + 8x^3 = 8 x^3$ and $0 x^6 + 5x^6 = 5x^6$.

Write the Final Answer

Now, combine all the terms to give the final answer. Starting with the highest power of $x$, this gives: $7 x^6 + 7 x^5 + 8 x^3 - 9 x^2 + 11 x - 5$.

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Key Concepts

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

Like Terms

When working with polynomials, it's essential to understand the concept of 'like terms.' Like terms are terms that contain the same variable raised to the same power. They can be combined because they fundamentally represent the same type of quantity.
Here are key points to remember about like terms:

Like terms have identical variable parts, including the exponents.
For example, in $7x^2 + 3x^2$, both terms are like terms because they share the variable $x$ raised to the power of $2$.
In the given exercise, terms like $7x^6$ and $0x^6$ are considered like terms because they have the same degree of $x^6$.
It's crucial to identify these terms first to facilitate the simplification process.

Recognizing like terms allows you to efficiently combine them, streamlining the process and making polynomial expressions simpler.

Polynomial Operations

Polynomial operations involve methods like addition, subtraction, and multiplication, applied to polynomials. In this exercise, we are focused on polynomial addition, which is an essential operation.
Here's how it works:

When adding polynomials, you align terms of the same degree or those considered like terms.
The terms are added by summing up their coefficients while maintaining the variable part unchanged.
For example, adding $2x^3$ and $5x^3$ gives you $7x^3$ since you merely add the coefficients $2$ and $5$.
Unmatched terms from polynomials, like $8x^3$ in our exercise, need to remain in the final expression, as there's no counterpart to combine with.

Understanding these operations can make complex expressions manageable and pave the way for solving higher-level polynomial equations.

Combining Like Terms

After identifying like terms, the next step is to combine them. This step is where you perform the actual operation of adding the coefficients of like terms together.
Combining like terms simplifies polynomials and involves a straightforward process:

Add or subtract the coefficients of like terms.
Ensure the variable and its exponent remain the same.
For example, $3x^2$ and $4x^2$ combine to $7x^2$ by adding their coefficients, $3 + 4$.
In $9x$ and $2x$, combining yields $11x$, simplifying the expression.

Remember, the goal is to reduce the polynomial expression to its simplest form by appropriately combining all possible like terms. This makes it easier to understand and work with further in any mathematical context.

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91影视

\(\left(7 x^6+2 x^5-3 x^2+9 x\right)+\left(5 x^5+8 x^3-6 x^2+2 x-5\right)\)

Short Answer

Step by step solution

Identify Like Terms

Add Like Terms

Write the Final Answer

Key Concepts

Like Terms

Polynomial Operations

Combining Like Terms

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