Problem 7 - Intermediate Algebra

Question

Suppose that after dividing \(2 x^{3}+5 x^{2}-11 x+4\) by \(2 x-1\) you obtain \(x^{2}+3 x-4\). Show how multiplication can be used to check the result.

Step-by-Step Solution

Answer

The multiplication confirms that the division is correct.

Step 1: Set up the Polynomial Expression

To verify the division, you need to multiply the quotient, which is \(x^2 + 3x - 4\), by the divisor, \(2x - 1\). Set up this expression for multiplication: \((x^2 + 3x - 4)(2x - 1)\).

Step 2: Distribute the Terms

Apply the distributive property. First, multiply \(x^2\) by each term in \(2x - 1\): \[(x^2)(2x) = 2x^3, \quad (x^2)(-1) = -x^2.\] Second, multiply \(3x\) by each term in \(2x - 1\): \[(3x)(2x) = 6x^2, \quad (3x)(-1) = -3x.\] Finally, multiply \(-4\) by each term in \(2x - 1\): \[(-4)(2x) = -8x, \quad (-4)(-1) = 4.\]

Step 3: Combine Like Terms

Add all the resulting terms from Step 2: \[2x^3, \ -x^2 + 6x^2, \ -3x - 8x, \ 4.\] Simplify by combining like terms: \[2x^3 + 5x^2 - 11x + 4.\]

Step 4: Verify the Result

Compare the result from multiplying \((x^2 + 3x - 4)(2x - 1)\) with the original polynomial, \(2x^3 + 5x^2 - 11x + 4\). Since they match, the multiplication confirms the correctness of the original division.

Key Concepts

Multiplication Verification

When it comes to polynomial division, a great way to confirm if your division process was correct is through multiplication verification. By multiplying the divisor and the quotient, you see if you can re-obtain the original polynomial. It's like using your result to work backward and check your math.

Here's how you do it:

• Set up the multiplication of the quotient and the divisor.
• Multiply every term in one polynomial by every term in the other.
• Collect all the terms you get and simplify them.
• Compare your result to the original polynomial to see if they match.

This multiplication method is not just a tool for checks and balances—it's a confidence booster! Once you master this, you'll have a reliable way to check your division answers, keeping errors at bay.

Distributive Property

The distributive property is a major player in making multiplication verification possible. It allows you to distribute each term of one polynomial across another, helping you multiply expressions systematically.

Here's how to apply it:

• Take each term from the first polynomial and multiply it with every term of the second polynomial.
• Write down these products as new terms.
• Ensure every term from both polynomials has been multiplied with each other, using parentheses to organize if needed.

This property isn't just about repeating multiplication processes; it's about organizing what's often complex into manageable parts. By breaking down the multiplication into smaller pieces, you can keep track of each contribution to the final result. And, as we saw, it's very useful for checking your polynomial division answers!

Combining Like Terms

Like grouping similar socks when you do laundry, combining like terms is all about bringing together similar elements to simplify your expression. Each time you multiply terms, you end up with several like terms that can be added together.

Here's what to remember:

• Identify terms that have the same variable raised to the same power. These are your like terms.
• Add or subtract these similar terms to combine them into a single term.
• Simplify the expression into a cleaner, final form.

Combining like terms is the icing on the cake in polynomial operations, allowing you to streamline your expressions. This is crucial, especially in multiplication verification, where combining accurately helps ensure your final expression matches the original polynomial used in your division problem. It's both the challenge and the reward of algebra!