Problem 88 Simplify each expression. $$ ... [FREE SOLUTION]

Chapter 1: Problem 88

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

Short Answer

Expert verified

$-7x^2 + 14$

Step by step solution

Distribute the Constants

Apply the distributive property to remove the parentheses. For the first term: \[ 2 \times (-2x^2 + 1) = -4x^2 + 2 \] For the second term: \[ -4 \times (x^2 - 3) = -4x^2 + 12 \]So, the expression becomes: \[ -4x^2 + 2 - 4x^2 + 12 + x^2 \]

Combine Like Terms

Simplify the expression by combining like terms. Combine the terms with $x^2$: \[ -4x^2 - 4x^2 + x^2 = -7x^2 \] Then combine the constant terms: \[ 2 + 12 = 14 \]Thus, the expression becomes: \[ -7x^2 + 14 \]

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Key Concepts

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

distributive property

Understanding the distributive property is essential for simplifying algebraic expressions. The distributive property states that multiplying a sum by a number is the same as doing each multiplication separately. For example: if you have the expression $a(b + c)$, it is equivalent to $ab + ac$.

In the exercise, we applied the distributive property to remove the parentheses in two terms:

First term: $2(-2x^2 + 1) = -4x^2 + 2$
Second term: $-4(x^2 - 3) = -4x^2 + 12$

By distributing the constants (2 and -4), we simplified the expressions within the parentheses.

combining like terms

Combining like terms involves adding or subtracting terms that have the same variable raised to the same power. This step helps in further simplifying the expression.

In our simplified expression $-4x^2 + 2 - 4x^2 + 12 + x^2$, we combined the $x^2$ terms and the constant terms separately:

Combine $x^2$ terms: $-4x^2 - 4x^2 + x^2 = -7x^2$
Combine constants: $2 + 12 = 14$

After combining the like terms, the simplified form of the expression becomes $-7x^2 + 14$.

polynomials

Polynomials are algebraic expressions that involve sums and differences of terms, called monomials. These terms can have variables raised to different powers with corresponding coefficients.

For example, in the expression $-7x^2 + 14$, we have:

$-7x^2$: a monomial with variable $x$ raised to the power of 2 and coefficient -7
$14$: a constant monomial, also called a constant term

Understanding how to simplify polynomials by using the distributive property and combining like terms is crucial for many algebra problems. Polynomials are foundational to many areas of mathematics and are used to express a wide range of functions.

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91影视

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

Short Answer

Step by step solution

Distribute the Constants

Combine Like Terms

Key Concepts

distributive property

combining like terms

polynomials

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