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Chapter 4: Problem 93

Subtract. $$ \begin{aligned} &5 t^{2}+2 t-6\\\ &5 t^{2}-3 t-9 \end{aligned} $$

Short Answer

Expert verified
5t + 3

Step by step solution

01

- Write down the polynomials

Write the given polynomials one below the other, aligned by like terms for clarity. The polynomials are:\[5t^2 + 2t - 6\]\[5t^2 - 3t - 9\]
02

- Distribute the negative sign

Subtracting polynomials involves distributing the negative sign to each term in the second polynomial. This gives:\[ -(5t^2 - 3t - 9) = -5t^2 + 3t + 9 \]
03

- Combine like terms

Combine the like terms from both polynomials:\[(5t^2 + 2t - 6) + (-5t^2 + 3t + 9)\]First, combine the \(t^2\) terms:\[5t^2 - 5t^2 = 0\]Next, combine the \(t\) terms:\[2t + 3t = 5t\]Finally, combine the constant terms:\[-6 + 9 = 3\]
04

- Write the result

After combining all like terms, write the resulting polynomial:\[5t\ + 3\]

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Key Concepts

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

Aligning Like Terms
To subtract polynomials, the first crucial step is aligning the like terms. Like terms are terms that have the same variable raised to the same power. For instance, in the polynomials given, both have terms with the variable t raised to the 2nd power, terms with t raised to the 1st power, and constant terms. By arranging these polynomials one under the other, we make it easier to subtract corresponding terms. So, let鈥檚 align them:
  • 5t虏 + 2t - 6
  • 5t虏 - 3t - 9
This organization will help simplify the next steps of the subtraction process.
Distributing Negative Sign
When subtracting one polynomial from another, we need to distribute the negative sign (or minus sign) across all the terms of the polynomial being subtracted. It鈥檚 like multiplying every term of the second polynomial by -1. Let鈥檚 see the steps:
  • Original polynomials:5t虏 + 2t - 65t虏 - 3t - 9
  • Apply the negative sign to the second polynomial:
    • -(5t虏 - 3t - 9)
    • which gives: -5t虏 + 3t + 9
    Distributing the negative sign is important because it ensures that each term in the polynomial being subtracted is correctly accounted for. Only once this step is done correctly can you move on to combining like terms.
Combining Like Terms
After aligning and distributing the negative sign, the final step is to combine like terms. Like terms are combined by adding or subtracting their coefficients. Let鈥檚 follow through with our example:
  • Combine the t虏 terms: 5t虏 - 5t虏 = 0
  • Combine the t terms:2t + 3t = 5t
  • Combine the constant terms:-6 + 9 = 3
After combining all like terms, the resultant polynomial is:5t + 3By following these steps, you ensure an accurate subtraction of polynomials. The steps are simple but crucial in arriving at the correct result.

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