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Chapter 10: Problem 46

Subtract the polynomials. $$\left(12 c^{2}+6 c-3\right)-\left(4 c^{2}-3 c+8\right)$$

Short Answer

Expert verified
The simplified polynomial is \( 8c^2 + 9c - 11\).

Step by step solution

01

- Write Down the Polynomials

Start by writing the given polynomials clearly: \ \(12c^2 + 6c - 3\) and \ \(4c^2 - 3c + 8\).
02

- Distribute the Minus Sign

Distribute the minus sign to each term in the second polynomial: \ \((4c^2 - 3c + 8)\) becomes \ \(-4c^2 + 3c - 8\).
03

- Combine Like Terms

Now, combine the like terms from the two polynomials: \ \( (12c^2 + 6c - 3) - (4c^2 - 3c + 8)\) \ becomes \ \( 12c^2 + 6c - 3 - 4c^2 + 3c - 8\).
04

- Simplify the Expression

Combine the like terms to simplify the expression: \ \( (12c^2 - 4c^2) + (6c + 3c) + (-3 - 8)\). This simplifies to: \ \( 8c^2 + 9c - 11\).

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Key Concepts

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

Polynomial Subtraction
When subtracting polynomials, remember that you are dealing with expressions that contain multiple terms. Each term is composed of coefficients and variables raised to certain powers. In the given exercise, you are asked to subtract one polynomial from another:
\(\big(12c^2 + 6c - 3\big) - \big(4c^2 - 3c + 8\big)\).
  • The first polynomial has three terms: \(12c^2, 6c, \text{and} -3\).
  • The second polynomial also has three terms: \(4c^2, -3c, \text{and} 8\).

To subtract one polynomial from another, you subtract each corresponding term. However, be mindful of the 'minus' sign!
Distributing the Negative Sign
One critical step in polynomial subtraction is distributing the negative sign across the second polynomial. This means you need to change the sign of each term in the polynomial being subtracted.
In the exercise:
\((4c^2 - 3c + 8)\),
when you distribute the minus sign, it becomes: \(-4c^2 + 3c - 8\). Every term in the second polynomial changes its sign. Distributing the negative sign correctly is crucial because it affects the subsequent steps.
  • Positive 4\(c^2\) becomes negative 4\(c^2\).
  • Negative 3c becomes positive 3c.
  • Positive 8 becomes negative 8.

This adjustment ensures you accurately represent the difference between the two polynomials.
Combining Like Terms
After distributing the negative sign, the next step is to combine like terms. Like terms are terms in the polynomial that have the same variables raised to the same power.
In our example, after distributing the negative sign, our expression looks like this: \(12c^2 + 6c - 3 - 4c^2 + 3c - 8\).
  • Combine the \$c^2\ terms: \(12c^2 - 4c^2 = 8c^2\)
  • Combine the linear \(c\) terms: \(6c + 3c = 9c\)
  • Combine the constant terms: \(-3 - 8 = -11\)

After combining like terms, the simplified polynomial is: \({8c^2 + 9c - 11}\). This resulting polynomial is the answer. Remember, carefully combine each set of like terms to arrive at the correct solution.

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