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Chapter 5: Problem 65

Subtract the polynomials. $$\left(6 y^{3}+2 y^{2}-y-11\right)-\left(y^{2}-8 y+9\right)$$

Short Answer

Expert verified
The result of subtracting the given polynomials is \(6y^{3} + y^{2} + 7y - 20\).

Step by step solution

01

Distribute Negative Sign

Apply the negative sign from the subtraction to all terms in the second polynomial. This changes the signs of all terms in the second polynomial. Then, the expression becomes: \[6y^{3} + 2y^{2} - y - 11 - y^{2} + 8y - 9\]
02

Combine Like Terms

The next step is to combine like terms. The terms that are the same are those with the same variables raised to the same power. Thus,\[6y^{3} + (2y^{2} - y^{2}) + (- y + 8y) + (-11 - 9)\] Simplify by combining the terms:\[6y^{3} + y^{2} + 7y - 20\]

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Key Concepts

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

Distribute Negative Sign
When subtracting polynomials, distributing the negative sign is crucial. This means that you must apply the negative sign to each term in the polynomial that's being subtracted. For example:
  • Original expression: \( (6y^3 + 2y^2 - y - 11) - (y^2 - 8y + 9) \)
  • Apply the negative sign to each term in the second polynomial: \(-y^2 + 8y - 9\)
After this step, it is as if you are adding the opposite of the second polynomial:\[ 6y^3 + 2y^2 - y - 11 - y^2 + 8y - 9 \]Remember:
  • A positive number becomes negative.
  • A negative number becomes positive.
This changes the signs and is necessary before combining like terms.
Combine Like Terms
Once you've adjusted the signs through distributing, it's time to combine like terms. This involves bringing together terms that have identical variables and exponents. For example:
  • Combine the \(y^2\) terms: \(2y^2\) and \(-y^2\) become \(y^2\).
  • Combine the \(y\) terms: \(-y\) and \(8y\) turn into \(7y\).
  • Combine the constant terms: \(-11 - 9\) results in \(-20\).
These calculations bring the expression together:\[ 6y^3 + y^2 + 7y - 20 \]Combining like terms simplifies the expression, making it easier to read and use.
Algebra
Algebra involves working with expressions that contain variables and constants. It often requires operations like addition, subtraction, multiplication, and division in combination with polynomials. In this instance, subtraction was used on polynomial expressions. To successfully solve algebraic problems involving polynomials:
  • Understand the order of operations: deal with signs first.
  • Manage terms carefully by pairing them by their degree (or power).
  • Use algebraic principles to simplify and solve equations.
Polynomials, which consist of several terms combined, can vary greatly in complexity, but with a steady approach using these principles, they become more manageable. Learning to manipulate polynomial expressions is a fundamental algebraic skill that extends to many higher-level math topics.

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$4^{-2}<4^{-3}$$

Perform the indicated operations. $$(y+1)\left(y^{2}-y+1\right)-(y-1)\left(y^{2}+y+1\right)$$

In Exercises \(79-82,\) simplify each expression. $$\frac{6 x^{3}(3 x-1)+5 x^{2}(6 x-3)}{3 x^{2}}$$

Will help you prepare for the material covered in the next section. In each exercise, find the indicated products. Then, if possible, state a fast method for finding these products. (You may already be familiar with some of these methods from a high school algebra course.) a. \((x+3)(x+4)\) b. \((x+5)(x+20)\)

Explain how to multiply monomials. Give an example.
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