Problem 510 Find the sum of $8 q^{3}-27$ a... [FREE SOLUTION]

91影视

Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 510

Find the sum of $8 q^{3}-27$ and $q^{2}+6 q-2$

Short Answer

Expert verified

The sum is $8q^{3} + q^{2} + 6q - 29$.

Step by step solution

Write Down the Given Expressions

The expressions to be added are (1) 8q^{3} - 27 (2) q^{2} + 6q - 2

Align the Expressions

Arrange one expression beneath the other, aligning the similar terms:\[\begin{array}{r} 8 q^{3} - 27 \+ q^{2} + 6 q - 2 \end{array}\]

Add the Like Terms

Combining the like terms results in:\[ 8 q^{3} + q^{2} + 6 q - 29 \]

Simplify the Expression

Ensure all like terms are combined properly to give: $8q^{3} + q^{2} + 6q - 29$

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Key Concepts

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

polynomial addition

Polynomial addition is all about adding two or more polynomials together. Polynomials are algebraic expressions composed of variables and coefficients, structured in terms of powers. When adding polynomials, the key idea is to add coefficients of terms with the same degree or power. This means you add the numbers in front of similar terms, not the exponents themselves. For example, in our given equations, the term with the highest degree in both polynomials is aligned first: $8q^{3}$. Whenever two polynomials need to be added, first write them down in a structured format where similar terms line up vertically.

like terms

The concept of 'like terms' is crucial when working with polynomial addition. Like terms have the same variable raised to the same power. For instance, in our given problem, there is no like term for $8q^{3}$ in the second polynomial, so it remains as $8q^{3}$. However, the $q^{2}$ term in the second polynomial aligns similarly with $q^{2}$ in the final answer. The numerical coefficients in front of these like terms are what you combine during polynomial addition. Make sure to treat constants (without variables) as like terms too, as shown for $-27$ and $-2$ in our example.

simplifying expressions

Simplifying expressions involves combining like terms to make the expression as concise as possible. After aligning the polynomials and combining the coefficients of the like terms, you simplify the resulting expression by ensuring no further like terms can be combined. In our example, we align and add the polynomials to get $8q^{3} + q^{2} + 6q - 29$. Each term is unique, and no further simplification is possible. Simplification reduces clutter and makes the polynomial more user-friendly, which is vital in algebraic problem-solving.

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91影视

Find the sum of \(8 q^{3}-27\) and \(q^{2}+6 q-2\)

Short Answer

Step by step solution

Write Down the Given Expressions

Align the Expressions

Add the Like Terms

Simplify the Expression

Key Concepts

polynomial addition

like terms

simplifying expressions

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