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Chapter 5: Problem 93

Simplify. $$ 4\left(x^{2}-x+3\right)-2\left(2 x^{2}+x-1\right) $$

Short Answer

Expert verified
-6x + 14

Step by step solution

01

Distribute the constants

Distribute the constants outside the parentheses to each term inside the parentheses. Apply distribution for both parts of the expression: 4(x^2 - x + 3) becomes 4*x^2 - 4*x + 4*3 -2(2x^2 + x - 1) becomes -2*2x^2 - 2*x - 2*(-1).
02

Simplify distributed terms

Simplify the distributed terms: 4*x^2 - 4*x + 4*3 = 4x^2 - 4x + 12 -2*2x^2 - 2*x - 2*(-1) = -4x^2 - 2x + 2.
03

Combine like terms

Combine the like terms from both expressions: 4x^2 - 4x + 12 - 4x^2 - 2x + 2. Combine the x^2 terms: 4x^2 - 4x^2 = 0 Combine the x terms: -4x - 2x = -6x Combine the constant terms: 12 + 2 = 14. This results in: -6x + 14.

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Key Concepts

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

distributive property
The distributive property is a key algebraic concept. It allows you to multiply a number by a group of terms inside parentheses. This property is essential for simplifying expressions and solving equations. The distributive property states: a(b + c) = ab + acWhen you apply it, you distribute (or multiply) the outside term to each inside term. For example, distribute in 4(x^2 - x + 3) and -2(2x^2 + x - 1) as shown:
  • 4(x^2 - x + 3) becomes 4*x^2 - 4*x + 4*3
  • -2(2x^2 + x - 1) becomes -2*2x^2 - 2*x - 2*(-1)
After distribution, your expressions are expanded and easier to simplify.
combining like terms
Combining like terms is another essential concept. It simplifies algebraic expressions and makes them more manageable. Like terms are terms that have the same variable raised to the same power, but they can have different coefficients. For instance, in the expression 4x^2 - 4x + 12 - 4x^2 - 2x + 2, the like terms are:
  • 4x^2 and -4x^2
  • -4x and -2x
  • 12 and 2
To combine like terms, add or subtract their coefficients:
  • 4x^2 - 4x^2 = 0x^2 (or just 0)
  • -4x - 2x = -6x
  • 12 + 2 = 14
So, combining like terms results in -6x + 14.
polynomials
Polynomials are expressions made up of terms that include variables raised to whole number powers and coefficients. Understanding polynomials is crucial for working with algebraic expressions. A polynomial can have constants, variables, and exponents, but the exponents must be whole numbers.
  • A polynomial in one variable, like 4x^2 - 4x + 12, is called a univariate polynomial.
  • Terms like 4x^2 and -4x are part of a polynomial.
  • The degree of a polynomial is the highest power of the variable in the polynomial. For example, the degree of 4x^2 - x + 3 is 2.
Polynomials can be added, subtracted, multiplied, and even divided in some cases. Simplifying polynomials involves distributing terms and combining like terms, just as we've done in our exercise.

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