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Chapter 6: Problem 53

Find the sum of \(\left(8 a^{3}-8 a\right)\) and \(\left(a^{2}+6 a+12\right)\).

Short Answer

Expert verified
The sum is \(8a^3 + a^2 - 2a + 12\).

Step by step solution

01

- Write the Expression

Identify the two expressions to be summed: \(8a^3 - 8a\) and \(a^2 + 6a + 12\).
02

- Combine the Expressions

Add together the two expressions: \(8a^3 - 8a + a^2 + 6a + 12\).
03

- Combine Like Terms

Group together like terms in the combined expression: \(8a^3 + a^2 - 8a + 6a + 12\).
04

- Simplify the Expression

Simplify the like terms in the expression to get the final result. Combine \(-8a\) and \(6a\) to get \(-2a\): \(8a^3 + a^2 - 2a + 12\).

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Key Concepts

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

Polynomials
Polynomials are algebraic expressions that include coefficients and variables. The variables have non-negative integer exponents. For example, in the given exercise, we have two polynomials: \(8a^3 - 8a\) and \(a^2 + 6a + 12\). Each term in these expressions is a monomial.

Polynomials are classified based on their degrees - the highest exponent of the variable. In \(8a^3 - 8a\), the highest degree is 3, making it a cubic polynomial. For \(a^2 + 6a + 12\), the highest degree is 2, so it is a quadratic polynomial.
Like Terms
Like terms are terms that have identical variable parts raised to the same power. They can be combined by adding or subtracting their coefficients.

For instance, in the expression \(8a^3 - 8a + a^2 + 6a + 12\), the terms \(8a^3\) and \(a^2\) are not like terms, but \(-8a\) and \(6a\) are like terms because both include the variable \(a\) to the first power.

Grouping like terms is essential in simplifying expressions. By recognizing and combining like terms, we can simplify the expression more easily, resulting in fewer terms and a less complex expression.
Simplification
Simplification means reducing an algebraic expression to its simplest form. This involves combining like terms and simplifying constants. Let's look at the exercise step-by-step:

First, we write the expression to be simplified: \(8a^3 - 8a + a^2 + 6a + 12\).

Next, we combine the like terms, starting with the \(-8a\) and \(6a\). Adding these gives \(-2a\).

Finally, we write the simplified expression: \(8a^3 + a^2 - 2a + 12\).

This new expression is easier to work with and effectively communicates the combined result of the original polynomials.

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Most popular questions from this chapter

Mixed Practice (a) \(q^{5} \cdot q^{3}\) (b) \(\left(q^{5}\right)^{3}\)

Divide each polynomial by the binomial. $$\left(3 b^{3}+b^{2}+2\right) \div(b+1)$$

Divide each polynomial by the monomial. $$\frac{51 y^{4}+42 y^{2}}{3 y^{2}}$$

Divide each polynomial by the monomial. $$\left(16 y^{2}-20 y\right) \div 4 y$$

Mixed Practice (a) \(p^{4} \cdot p^{6}\) (b) \(\left(p^{4}\right)^{6}\)
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