Problem 55 - Algebra A Combined Function

Question

Subtract \(\left(3 x^{2}-4\right)\) from the sum of \(\left(x^{2}-9 x+2\right)\) and \(\left(2 x^{2}-6 x+1\right)\)

Step-by-Step Solution

Answer

The result of the operation is \(-15x + 7\).

Step 1: Identify the Expressions

We need to subtract the polynomial \( (3x^2 - 4) \) from the sum of \( (x^2 - 9x + 2) \) and \( (2x^2 - 6x + 1) \). First, identify these three expressions.

Step 2: Add the Two Given Polynomials

Add the polynomials \( (x^2 - 9x + 2) \) and \( (2x^2 - 6x + 1) \). Combine like terms, giving us \(\begin{align*} (x^2 - 9x + 2) + (2x^2 - 6x + 1) &= x^2 + 2x^2 - 9x - 6x + 2 + 1 \ &= 3x^2 - 15x + 3. \end{align*}\)

Step 3: Set up the Subtraction

We now subtract the polynomial \( (3x^2 - 4) \) from the result found in Step 2: \( (3x^2 - 15x + 3) \). This gives us the expression \( (3x^2 - 15x + 3) - (3x^2 - 4) \).

Step 4: Perform the Subtraction

Subtract the two expressions by distributing the negative sign and combining like terms:\[(3x^2 - 15x + 3) - (3x^2 - 4) = 3x^2 - 15x + 3 - 3x^2 + 4 = -15x + 7.\]

Step 5: Verify the Solution

Check each term to ensure proper subtraction:- The \(3x^2\) terms cancel each other out.- The \(-15x\) remains unchanged as there are no terms to combine with.- The constant term becomes \(3 + 4 = 7\). Thus, the final polynomial is \(-15x + 7\).

Key Concepts

Polynomial Addition

Polynomial addition is the process of summing two or more polynomials to create a new polynomial. To effectively add polynomials, you need to:

• Identify each polynomial involved in the addition. In our case, these were \((x^2 - 9x + 2)\) and \((2x^2 - 6x + 1)\).
• Combine the terms of the same degree, which we call "like terms". For example, terms like \(x^2\) from both expressions can be added together.

Start by arranging all terms in a line, keeping the like terms aligned. Then, sum them up straightforwardly:\[(x^2 - 9x + 2) + (2x^2 - 6x + 1) = 3x^2 - 15x + 3.\]This condensed form neatly consolidates the original polynomials into a simplified expression.

Like Terms

Like terms in polynomials are the terms that have the exact same variable raised to the same power. Understanding and identifying like terms is essential for both addition and subtraction of polynomials. Here's how you can recognize them:

• Check for terms with the same variable and exponent. For example, \(x^2\) terms can only be combined with other \(x^2\) terms.
• Constant terms, like numbers without variables, can be combined with other constants.

During our addition process, we identified and combined like terms from the polynomials \((x^2 - 9x + 2)\) and \((2x^2 - 6x + 1)\):- The \(x^2\) terms: \(x^2 + 2x^2 = 3x^2\)- The \(x\) terms: \(-9x - 6x = -15x\)- The constant terms: \(2 + 1 = 3\)This simplification aligns terms in a way that makes further calculations more manageable.

Distributive Property

The distributive property is a useful algebraic principle that applies to numbers and variables alike. It states that when you multiply a sum by a number, you can distribute the multiplication to each addend separately. This concept is key during the subtraction of polynomials, particularly when managing negative signs.Here's how we used it in our polynomial subtraction:We had \[(3x^2 - 15x + 3) - (3x^2 - 4).\]To subtract, distribute the negative sign to each term in the second polynomial:\[- (3x^2 - 4) = -3x^2 + 4.\]By applying the distributive property, we simplify the calculation effectively, ultimately reaching our simplified expression of \(-15x + 7\).

Verify Solution

Verifying the solution is the final step to ensure calculations are accurate and correct. This involves checking each step to confirm that operations followed algebraic rules properly. In our exercise, here's what we did:

• Revisited the cancellation of like terms: \(3x^2 - 3x^2 = 0\). This confirms those terms correctly cancel each other out.
• Re-evaluated the remaining terms: \(-15x + 0 = -15x\).
• Confirmed the addition and subtraction of constant terms: \(3 - (-4) = 3 + 4 = 7\).

Reviewing each calculation verified that nothing was skipped or miscalculated, affirming the final solution \(-15x + 7\) as both complete and correct. Ensuring this reinforces understanding and builds confidence in performing polynomial operations accurately.