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Chapter 10: Problem 14

Perform each operation and combine like terms. a. \(\left(x^{2}+5 x-4\right)-\left(3 x^{3}-2 x^{2}+6\right)\) b. \((x+7)\left(x^{4}-4 x\right)\) c. \(3 x+7(x+y)-4 y(x-8)\)

Short Answer

Expert verified
a. \(-3x^3 + 3x^2 + 5x - 10\); b. \(x^5 + 7x^4 - 4x^2 - 28x\); c. \(10x + 39y - 4xy\)

Step by step solution

01

Distribute Minus Sign

First, distribute the minus sign across the second expression in part a. This means we change the signs of each term inside the brackets: \[ \left(x^{2} + 5x - 4\right) - \left(3x^{3} - 2x^{2} + 6\right) = x^{2} + 5x - 4 - 3x^{3} + 2x^{2} - 6 \]
02

Combine Like Terms for Part A

Now, combine like terms in the expression from Step 1:- The \(x^2\) terms: \(x^2 + 2x^2\) = \(3x^2\)- The constant terms: \(-4 - 6\) = \(-10\)The expression simplifies to:\[ -3x^{3} + 3x^{2} + 5x - 10 \]
03

Distribute for Part B

For part b, distribute the terms in \((x + 7)(x^4 - 4x)\):- \(x \cdot x^4 = x^5\)- \(x \cdot (-4x) = -4x^2\)- \(7 \cdot x^4 = 7x^4\)- \(7 \cdot (-4x) = -28x\)Combine these results:\[ x^5 + 7x^4 - 4x^2 - 28x \]
04

Simplify Expand and Combine Like Terms for Part C

First, distribute the 7 and -4y in part c:- \(7(x) = 7x\)- \(7(y) = 7y\)- \(-4y(x) = -4xy\)- \(-4y(-8) = 32y\)This gives us:\[ 3x + 7x + 7y - 4xy + 32y \]Now combine like terms:- \( 3x + 7x = 10x \)- \( 7y + 32y = 39y \)Resulting in:\[ 10x + 39y - 4xy \]
05

Final Combined Expression

For each sub-part, we performed operations and combined like terms:- **Part A:** \(-3x^3 + 3x^2 + 5x - 10 \)- **Part B:** \(x^5 + 7x^4 - 4x^2 - 28x \)- **Part C:** \(10x + 39y - 4xy \)These are the simplified expressions for each part of the problem.

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Key Concepts

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

Algebraic Expressions
Algebraic expressions are combinations of variables, constants, and operations like addition, subtraction, multiplication, and division. Variables like \(x\) and \(y\) represent unknown values that can change, making them crucial in algebra.
Understanding these expressions involves recognizing the distinct parts of each, such as coefficients (e.g., the 5 in \(5x\)), which multiply the variables, and constants (e.g., -4), which stand alone.
To solve problems involving algebraic expressions, one often needs to perform operations like addition and subtraction on terms. This requires careful handling of terms especially when it involves operations like distributing a minus sign or applying the distributive property.
Overall, mastering algebraic expressions is essential as they are the foundation for more complex mathematical problems.
Distributive Property
The distributive property is a fundamental concept in algebra that allows us to break down expressions into simpler parts. It's particularly useful when multiplying a single term by terms inside brackets, like in part b of the original exercise.
This property states that for all numbers \(a\), \(b\), and \(c\), we have: \(a(b + c) = ab + ac\). This means you multiply the single term \(a\) by each of the terms inside the parentheses and then sum them.
In practical terms, for expressions like \((x + 7)(x^4 - 4x)\), the distributive property helps break it down:
  • Multiply \(x\) by each term inside the parenthesis: \(x \cdot x^4\) and \(x \cdot (-4x)\).
  • Do the same with 7: \(7 \cdot x^4\) and \(7 \cdot (-4x)\).
By applying the distributive property, this expression becomes easier to simplify, as shown in the step-by-step solution.
Combining Like Terms
Combining like terms is a critical skill for simplifying algebraic expressions. It involves adding or subtracting terms that have the same variable raised to the same power.
Consider this approach like tidying up: group similar items together. For example, in an expression like \(x^2 + 2x^2 + 5x\), you would combine the \(x^2\) terms to get \(3x^2\).
When engaging in polynomial operations, especially after using the distributive property, it's crucial to scan through the expression to spot terms that can be combined. During Step 2 in Part A of the original exercise, terms like \(x^2 + 2x^2\) were combined to form a simpler expression \(3x^2\).
  • Identify terms with the same variable and power.
  • Add or subtract the numerical coefficients of these terms.
By combining like terms, expressions become more manageable and easier to work with, ultimately leading to a simpler and clearer solution.

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