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Chapter 2: Problem 41

Simplify each expression as completely as possible. $$3 x^{2}-7 x+4-8 x^{2}-3-x$$

Short Answer

Expert verified
-5x^2 - 8x + 1

Step by step solution

01

Combine Like Terms for the Quadratic Term

Identify and combine the quadratic terms in the expression. Here, the quadratic terms are \(3x^2\) and \(-8x^2\).\[3x^2 - 8x^2 = -5x^2\]
02

Combine Like Terms for the Linear Term

Identify and combine the linear terms in the expression. Here, the linear terms are \(-7x\) and \(-x\).\[-7x - x = -8x\]
03

Combine Like Terms for the Constant Term

Identify and combine the constant terms in the expression. Here, the constant terms are \(4\) and \(-3\).\[4 - 3 = 1\]
04

Write the Simplified Expression

Combine all the simplified terms to write the final simplified expression.\[-5x^2 - 8x + 1\]

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Key Concepts

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

Combining Like Terms
When we simplify algebraic expressions, combining like terms is an essential step. Like terms are terms that have the same variable raised to the same power. For example, in the expression, $$3x^2 - 7x + 4 - 8x^2 - 3 - x$$, we can see like terms which share similar variables and powers.
Quadratic Expressions
A quadratic expression includes terms where the variable is raised to the second power (e.g. \( x^2 \)). In the given expression, the quadratic terms are \( 3x^2 \) and \( -8x^2 \). When we combined these, we got: define the quadratic terms: \[ 3x^2 - 8x^2 = -5x^2 \]. This gives us the quadratic part of the simplified expression.
Linear Expressions
Linear expressions have terms where the variable is raised to the first power (e.g. \( x \)). In our example, the linear terms are \( -7x \) and \( -x \). Combining them looks like this \[ -7x - x = -8x \]. Adding the coefficients make for the linear part of our simplified term.
Constants in Algebra
Constants are terms that don鈥檛 have any variables attached. In our given expression, the constants are \( 4 \) and \( -3 \). When we combined these, we got: \[ 4 - 3 = 1 \]. This provides the constant part of the final expression. Combining all parts, we get our simplified expression: defined as, \[ -5x^2 - 8x + 1 \]

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