Problem 69 - Algebra 1: Concepts and Skills

Question

Add or subtract the polynomials. (Lesson 10.1) $$\left(-5 x^{2}+2 x-12\right)-\left(6-9 x-7 x^{2}\right)$$

Step-by-Step Solution

Answer

The solution to the given problem is \(2x^2 + 11x -18\).

Step 1: Distribute the negative sign to the second polynomial

Distribute the negative sign across the terms in the second polynomial to change its sign: \( -5x^2 + 2x - 12 - (6 - 9x - 7x^2) = -5x^2 + 2x - 12 - 6 + 9x + 7x^2 \)

Step 2: Rearrange the terms

Rearrange the terms to group the like terms together: \( (-5x^2 + 7x^2) + (2x + 9x) - (12 + 6) \)

Step 3: Combine like terms

Combine like terms by performing the operations: \(2x^2 + 11x -18\)

Key Concepts

Combining Like Terms

When working with polynomials, one crucial step is combining like terms. Like terms refer to terms in a polynomial that contain the same variable raised to the same power. This similarity is key because only like terms can be combined or added together to simplify the equation.

For instance, consider these terms:

• \(-5x^2\) and \(7x^2\) are like terms since they both have the variable \(x\) raised to the power 2.
• \(2x\) and \(9x\) are like terms as they both have the variable \(x\) raised to the power 1.

When combining like terms in the expression \((-5x^2 + 7x^2) + (2x + 9x)\), you perform the addition separately for each set of like terms:

• \(-5x^2 + 7x^2= 2x^2\)
• \(2x + 9x = 11x\)

By combining these, you condense the polynomial to \(2x^2 + 11x\). This makes the polynomial simpler and easier to work with, which is especially useful when performing further operations.

Distributing

Distributing involves multiplying a single term across the terms within parentheses. In subtraction or addition of polynomials, distribution often means applying a sign change to each term within a set of parentheses.

For example, in the problem \(-5x^2 + 2x - 12 - (6 - 9x - 7x^2)\), the negative sign before the parentheses means each term inside must be flipped in sign upon distributing over \((6 - 9x - 7x^2)\).

• Distribute the negative sign: \(-6\) becomes \(-6\) (effectively no change)
• The \(-(-9x)\) becomes \(+9x\) since two negatives make a positive.
• And lastly, \(-(-7x^2)\) becomes \(+7x^2\).

This distribution step is necessary to prepare the polynomial for further simplification, ensuring that the signs are correct for combining like terms effectively.

Subtraction of Polynomials

Subtracting polynomials involves distributing a negative sign through the polynomial you are subtracting and then combining like terms. This operation can be visualized as altering the subtraction process into an addition of adjusted terms.

Start by rewriting the expression with the second polynomial reversed in sign, which is what occurs in: \(-5x^2 + 2x - 12 - (6 - 9x - 7x^2)\). After distributing, the expression is \(-5x^2 + 2x - 12 - 6 + 9x + 7x^2\).

Once the signs are correctly adjusted, the expression becomes ready for combining like terms.

• This changes the polynomial expression from something potentially complicated to something easily manageable.
• The resulting polynomial: \(2x^2 + 11x - 18\) is neater, making analysis or further operations straightforward.

Subtraction, by mindfully distributing negatives and combining alike, becomes straightforward and efficient.