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Chapter 4: Problem 81

Perform the indicated operations. Subtract \(4 y^{2}-2 y+3\) from \(7 y^{2}-6 y+5\)

Short Answer

Expert verified
\(3y^2 - 4y + 2\)

Step by step solution

01

Write down the polynomials

Identify the polynomials involved in the operation:First polynomial: - Polynomial to subtract: \(4y^{2} - 2y + 3\)
02

Set up the subtraction

Write the operation in standard form:\((7y^2 - 6y + 5) - (4y^{2} - 2y + 3)\)
03

Distribute the negative sign

Distribute the subtraction sign across the second polynomial:\(7y^2 - 6y + 5 - 4y^{2} + 2y - 3\)
04

Combine like terms

Combine the like terms in the expression:\[(7y^2 - 4y^2) + (-6y + 2y) + (5 - 3)\]This simplifies to:\[3y^2 - 4y + 2\]

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Key Concepts

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

Distributing the Negative Sign
Distributing the negative sign correctly is crucial in polynomial subtraction. This is often where students make mistakes. Let's look at the operation step by step:
  • When you subtract one polynomial from another, you are essentially adding the inverse of the polynomial being subtracted.
  • In our case, we need to subtract \(4y^{2} - 2y + 3\) from \(7 y^{2} - 6 y + 5\).
  • This means we change each sign in \(4y^{2} - 2y + 3\).

So, \( - (4y^{2} - 2y + 3) \) becomes \( -4y^{2} + 2y - 3 \). The negative sign is distributed to every term within the parentheses: the positive terms become negative and the negative terms become positive. Make sure to carefully distribute the sign to avoid errors!
Combining Like Terms
Combining like terms is a key step in simplifying polynomials. Once you've distributed the negative sign, the next step is to combine the terms that have the same variable raised to the same power.

In our example: After distribution, we have \( 7y^2 - 6y + 5 - 4y^2 + 2y - 3 \).

To combine like terms:
  • Group the \(y^2\) terms: \( 7y^2 - 4y^2 \)
  • Group the \(y\) terms: \( -6y + 2y \)
  • Group the constant terms: \( 5 - 3 \)

Now, combine them by adding or subtracting the coefficients:
  • \(7y^2 - 4y^2 = 3y^2 \)

  • \( -6y + 2y = -4y \)

  • \( 5 - 3 = 2 \)

    • This yields:\br \( 3y^2 - 4y + 2 \).
Polynomial Operations
Polynomial operations might seem complex, but when you break them down into smaller steps, they become easier to handle.
  • Addition and subtraction of polynomials involve combining like terms.
  • Multiplication might involve distributing each term in one polynomial to every term in the other polynomial (which we did not cover here).
  • Keep the principles of distributing the negative sign and combining like terms in mind to ensure you perform polynomial subtraction correctly.

Once you master these crucial steps, you will be able to handle more complex polynomial operations with confidence. Practice makes perfect, so continually practicing these kinds of problems is beneficial!

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Most popular questions from this chapter

Write using only positive exponents and then evaluate. Assume that all variables represent nonzero real numbers. $$ \left(\frac{3 z}{4}\right)^{-3} $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \left(\frac{5 m^{4} n^{-3}}{m^{-5} n^{2}}\right)^{-2} $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \left(3 p^{-4}\right)^{2}\left(p^{3}\right)^{-1} $$

The factors in the following exercises involve fractions or decimals. Apply the methods of this section, and find each product. $$ (0.1 y+2.1)(0.5 y-0.4) $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \left(\frac{-4 a^{3} b^{2}}{12 a^{5} b^{-4}}\right)^{-3} $$
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