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Chapter 5: Problem 43

Subtract. $$ \left(1.2 x^{3}+4.5 x^{2}-3.8 x\right)-\left(-3.4 x^{3}-4.7 x^{2}+23\right) $$

Short Answer

Expert verified
4.6x^{3} + 9.2x^{2} - 3.8x - 23

Step by step solution

01

- Write Down the Expressions

Write down the given expressions to be subtracted: Expression 1: 1.2x^{3} + 4.5x^{2} - 3.8x Expression 2: -3.4x^{3} - 4.7x^{2} + 23
02

- Distribute the Negative Sign

Distribute the negative sign in front of the second expression across each term within the parentheses: -(-3.4x^{3}) -(-4.7x^{2}) -(23) This simplifies to: 3.4x^{3} + 4.7x^{2} - 23
03

- Combine Like Terms

Combine the like terms from both expressions: First, group and add/subtract the coefficients of the same-power terms: (1.2x^{3} + 3.4x^{3}) + (4.5x^{2} + 4.7x^{2}) + (-3.8x - 0) - 23 This simplifies to: (4.6x^{3}) + (9.2x^{2}) + (-3.8x)-23

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Key Concepts

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

Combining Like Terms
When dealing with polynomials, one of the crucial steps is combining like terms. Like terms are terms that have the same variable raised to the same power. For example, in the expression \(1.2x^{3} + 4.5x^{2} - 3.8x\), the terms with \(x^{3}\), \(x^{2}\), and \(x^{1}\) are like terms because they share the same variables and powers.
To combine like terms, follow these steps:
  • Identify the terms with the same variable and exponent.
  • Add or subtract the coefficients of these terms.
For instance, if we have \(1.2x^{3}\) and \(3.4x^{3}\), we combine them to get \((1.2 + 3.4)x^{3} = 4.6x^{3}\).
This process is essential because it simplifies the polynomial, making it easier to work with and understand.
Distributing Negative Signs
Distributing negative signs can often be a stumbling block when subtracting polynomials. It's important to reverse the signs of each term in the subtracted polynomial. In our example, we have the expression \(-(-3.4x^{3} - 4.7x^{2} + 23)\).
To distribute the negative sign, we change the signs of all the terms inside the parentheses:
  • The \(-(-3.4x^{3})\) becomes \(+3.4x^{3}\).
  • The \(-(-4.7x^{2})\) becomes \(+4.7x^{2}\).
  • The \(-(23)\) becomes \(-23\).
Now, the expression simplifies to \(3.4x^{3} + 4.7x^{2} - 23\).
This makes it easier to combine like terms in the next step.
Understanding Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations (such as addition and subtraction). In our problem, the expressions are:
  • Expression 1: \(1.2x^{3} + 4.5x^{2} - 3.8x\)
  • Expression 2: \(-3.4x^{3} - 4.7x^{2} + 23\)
Subtracting an algebraic expression involves distributing the negative sign and then combining like terms. This is a common and fundamental algebraic operation that forms the basis for simplifying complex mathematical problems.
Algebraic expressions allow us to represent real-world situations mathematically and solve for unknown values. By understanding how to work with these expressions, you can tackle a variety of problems in algebra and beyond.

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