Problem 93 - Elementary Algebra

Question

Problem: Simplify: \(\left(5 x^{2}-8\right)-(3 x-9)\) Incorrect Answer: \(\left(5 x^{2}-8\right)-(3 x-9)\) $$ \begin{aligned} &=5 x^{2}-8-3 x+9 \\ &=2 x^{2}+1 \end{aligned} $$

Step-by-Step Solution

Answer

5x^2 - 3x + 1

Step 1 - Distribute the negative sign

To simplify the expression \((5x^2 - 8) - (3x - 9)\), distribute the negative sign to each term inside the parentheses. This gives: \[5x^2 - 8 - 3x + 9\].

Step 2 - Combine like terms

Now, combine the like terms in the expression. The like terms are \(- 8 + 9\) and the remaining terms are \[5x^2 - 3x \], resulting in: \[5x^2 - 3x + 1\].

Key Concepts

Distributing the Negative Sign

When you encounter an expression with a negative sign in front of parentheses, you should distribute the negative sign to every term inside the parentheses. This changes the signs of those terms.

For example, in the problem \( \left(5 x^{2}-8\right)-\left(3 x-9\right) \), you would distribute the negative sign to the \(3x-9\) portion. This gives:
\[ 5x^2 - 8 - 3x + 9 \].

Notice how \(3x\) becomes \(-3x\) and \( -9 \) becomes \(+9\).

• If there's no coefficient in front of a variable or number, it's assumed to be 1. So, you multiply \(1 * -1 \) for each term in the parentheses.
• Distributing the negative sign correctly is crucial, otherwise, it can lead to completely wrong results.

Combining Like Terms

After distributing the negative sign, the next step in simplifying the expression is to combine like terms.

Like terms are terms that have the same variable raised to the same power.

• In our example: \[ 5x^2 - 8 - 3x + 9 \], \(5x^2\) and \(-3x \) both include terms with different variables.
  
  \( -8 \) and \(+9\) are constant terms (numbers without variables).
• Add or subtract the like terms: Constant terms \-8 + 9 = 1 \ gives us the final simplified expression.

Therefore, \[5x^2 - 3x + 1 \] is the simplified result.

Understanding Polynomial Expressions

Polynomial expressions are sums of terms, each consisting of a variable raised to a non-negative integer power, multiplied by a coefficient.

In our example: \[ 5x^2 - 3x + 1 \], this is a polynomial expression.

• The term '5x^2' has a coefficient of 5 and a variable \(x\) raised to the power of 2.
• \( -3x\) has a coefficient of -3 and a variable \( x \) raised to the power of 1.
• The constant term '1' has no variable associated with it, and it's considered a polynomial of degree 0.

By understanding how to manipulate polynomials through addition, subtraction, and distribution of factors, you can simplify even complex expressions efficiently.

Remember, polynomial expressions often show up in algebraic problems, so mastering these basics will help you immensely in your math journey.