Problem 39 - College Algebra with Modeling and Visualization

Question

Subtract the polynomials. $$\left(4 x^{4}+2 x^{2}-9\right)-\left(x^{4}-2 x^{2}-5\right)$$

Step-by-Step Solution

Answer

The result is \(3x^4 + 4x^2 - 4\).

Step 1: Distribute the Negative Sign

The expression needs to be simplified by distributing the negative sign across the second polynomial. Rewrite the expression as: \( 4x^4 + 2x^2 - 9 - x^4 + 2x^2 + 5 \).

Step 2: Combine Like Terms

Now, group and combine the like terms from the expression: \( (4x^4 - x^4) + (2x^2 + 2x^2) + (-9 + 5) \).

Step 3: Simplify Each Group

Perform the operations inside each group: - For \( 4x^4 - x^4 \), the result is \( 3x^4 \).- For \( 2x^2 + 2x^2 \), the result is \( 4x^2 \).- For \( -9 + 5 \), the result is \( -4 \).

Step 4: Write the Final Simplified Polynomial

Combine all the simplified terms to form the final polynomial: \( 3x^4 + 4x^2 - 4 \).

Key Concepts

Distributing Negative Sign

Distributing the negative sign across terms when subtracting polynomials is a crucial skill. Subtraction of polynomials isn't simply about removing terms; it's about changing their signs. Think of subtracting as adding the opposite. For example, when you encounter an expression like \[\left( 4x^4 + 2x^2 - 9 \right) - \left( x^4 - 2x^2 - 5 \right),\]you need to distribute the negative sign to each term in the second polynomial, effectively flipping their signs:

• \( x^4 \) becomes \(-x^4\)
• \(-2x^2 \) becomes \(+2x^2\)
• \(-5 \) becomes \(+5\)

This transforms the expression into:\[4x^4 + 2x^2 - 9 - x^4 + 2x^2 + 5\]This step ensures each term in the polynomial is correctly represented, setting the stage for accurate combination of like terms.

Combining Like Terms

Combining like terms is all about finding terms in a polynomial that share the same variable and exponent and then adding their coefficients together. This helps consolidate the expression into a simpler form. Consider the expression after distributing the negative sign:\[4x^4 + 2x^2 - 9 - x^4 + 2x^2 + 5\]Look for terms with the same variable degree and group them:

• \(4x^4\) and \(-x^4\)
• \(2x^2\) and \(2x^2\)
• \(-9\) and \(+5\)

Adding their coefficients gives:- \(4x^4 - x^4 = 3x^4\)- \(2x^2 + 2x^2 = 4x^2\)- \(-9 + 5 = -4\)This step organizes the expression into a neater form, making further simplification easier and more straightforward.

Simplification of Expressions

The final step in polynomial subtraction is simplification, where you finalize the expression by eliminating redundant terms and organizing it into a conventional format. After combining the like terms, you're left with:\[3x^4 + 4x^2 - 4\]This is the simplified version of the original polynomial expression. Simplified expressions are valuable because they are easier to understand, evaluate, and apply in further algebraic manipulations. It's important to ensure that all terms in the final expression are organized by descending order of their degrees. Simplification not only ensures mathematical accuracy, but also helps communicate solutions clearly in mathematics.