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Chapter 5: Problem 135

Subtract. \\[ \begin{array}{r} -3 a^{2}+4 a-5 \\ 5 a^{2}+3 a-9 \end{array} \\]

Short Answer

Expert verified
-8a^2 + a + 4

Step by step solution

01

Title - Write Down the Polynomials

First, write down the polynomials that need to be subtracted:\[(-3a^2 + 4a - 5) \]and\[(5a^2 + 3a - 9)\]
02

Title - Distribute the Negative Sign

Subtracting a polynomial is the same as adding its opposite. Distribute the negative sign to the second polynomial:\[(5a^2 + 3a - 9) \text{ becomes } (-5a^2 - 3a + 9)\]
03

Title - Combine Like Terms

Now combine like terms:\[(-3a^2 + 4a - 5) + (-5a^2 - 3a + 9)\]Grouping the terms, we get:\[(-3a^2 + -5a^2) + (4a + -3a) + (-5 + 9)\]
04

Title - Simplify the Expression

Simplify each group:\[(-3a^2 - 5a^2) = -8a^2\]\[(4a - 3a) = a\]\[(-5 + 9) = 4\]Thus, the polynomial simplifies to:\[-8a^2 + a + 4\]

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Key Concepts

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

like terms
When dealing with polynomial subtraction and addition, it's crucial to understand 'like terms.' Like terms are terms that have the same variable raised to the same power. Only the coefficients (the numbers in front of the variables) are different.
For example, in the expression \(-3a^2 \text{ and } 5a^2\), \(a^2\) is the like term. Similarly, \(4a\text{ and }3a\) are like terms because they both contain the variable \(a\) to the first power.
By focusing on like terms, we can combine and simplify polynomials more easily. Avoid pairing terms that aren't alike as this can lead to errors in calculations.
distributing negative sign
Subtracting a polynomial is often simplified by adding its opposite. This step requires us to 'distribute the negative sign' across each term in the polynomial.
Consider the expression: \((5a^2 + 3a - 9)\). When distributing the negative sign, each term inside the parentheses changes its sign. This converts the expression to \((-5a^2 - 3a + 9)\).
Think of it as multiplying each term by -1. This straightforward step is essential for combining the polynomials correctly.
combining polynomials
Once we have our two polynomials with correctly distributed signs, the next step is combining them. In this case, we also need to focus on grouping like terms.
For our example:\[(-3a^2 + 4a - 5) + (-5a^2 - 3a + 9)\]We group like terms as follows:
  • Combine \(-3a^2\text{ and }-5a^2\)
  • Combine \(4a\text{ and }-3a\)
  • Combine \(-5\text{ and }9\)
This gives us groups that are easier to handle. Recall that grouping helps visualize the terms and ensures no term is overlooked.
simplifying expressions
After combining like terms, the final step is simplifying. Simplifying means performing arithmetic operations for each group.
Looking at our example, we handle each group as:
  • \((-3a^2 - 5a^2) = -8a^2\)
  • \((4a - 3a) = a\)
  • \((-5 + 9) = 4\)

Putting it all together, the simplified expression is \(-8a^2 + a + 4\). This is our final solution. Simplifying expressions helps in reducing them to a more manageable and understandable form.

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Most popular questions from this chapter

Write each number in scientific notation. $$ 0.0072 $$

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