Problem 627 Simplify each expression. $$\l... [FREE SOLUTION]

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Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 627

Simplify each expression. $$\left(5 a^{2}+2 a-12\right)+\left(9 a^{2}+8 a-4\right)$$

Short Answer

Expert verified

14a^2 + 10a - 16

Step by step solution

Identify like terms

Identify the terms with the same variable and exponent in each polynomial. For this problem, group the terms: $\text{Terms with } a^2: 5a^2 \text{ and } 9a^2$ $\text{Terms with } a: 2a \text{ and } 8a$ $\text{Constant terms: } -12 \text{ and } -4$

Combine the like terms

Add the coefficients of the identified like terms: $5a^2 + 9a^2 = 14a^2$ $2a + 8a = 10a$ $-12 + (-4) = -16$

Write the simplified expression

Combine the results from Step 2 to form the simplified expression: $14a^2 + 10a - 16$

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Key Concepts

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

Like Terms

When simplifying polynomials, understanding 'like terms' is essential.
'Like terms' are terms in a polynomial that have the same variable raised to the same power.
For example:

\term: $5a^2$ and $9a^2$ are like terms because both have 'a' raised to the power of 2.
Similarly, $2a$ and $8a$ are like terms as they both have 'a' raised to the power of 1.
Constants like -12 and -4 are also like terms since they don't contain variables.

Identifying like terms is the first step in simplifying polynomials.
Grouping them is crucial for the next step: combining their coefficients.

Combining Coefficients

After identifying like terms, the next step is 'combining coefficients'.
A coefficient is the numerical part of a term.
For instance, in $5a^2$, 5 is the coefficient.
To combine coefficients, simply add or subtract them as per the algebraic operation involved.
Let's see how it's done:

For the like terms $5a^2$ and $9a^2$: Add coefficients -> $5 + 9 = 14$. So, $5a^2 + 9a^2 = 14a^2$.
For the like terms $2a$ and $8a$: Add coefficients -> $2 + 8 = 10$. Thus, $2a + 8a = 10a$.
For constants -12 and -4: Add them -> $-12 + (-4) = -16$. Therefore, $-12 + (-4) = -16$.

Combining coefficients simplifies each group of like terms, preparing us to write the final, simplified expression.

Polynomial Addition

Polynomial addition involves combining like terms and their coefficients.
Here, we add two polynomials: $\big(5a^2 + 2a - 12\big)$ and $\big(9a^2 + 8a - 4\big)$.
Use what we've learned about like terms and combining coefficients:

First, group like terms: $5a^2$ with $9a^2$, $2a$ with $8a$, -12 with -4.
Next, add the coefficients: $5a^2 + 9a^2 = 14a^2$, $2a + 8a = 10a$, $-12 + (-4) = -16$.

Finally, write the simplified expression: $14a^2 + 10a - 16$.
By following these steps, you can easily simplify polynomials using polynomial addition.

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91影视

Simplify each expression. $$\left(5 a^{2}+2 a-12\right)+\left(9 a^{2}+8 a-4\right)$$

Short Answer

Step by step solution

Identify like terms

Combine the like terms

Write the simplified expression

Key Concepts

Like Terms

Combining Coefficients

Polynomial Addition

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