Problem 41 - Algebra 2

Question

Simplify. Classify each result by number of terms. $$ \left(7 x^{3}+9 x^{2}-8 x+11\right)-\left(5 x^{3}-13 x-16\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(2x^3 + 9x^2 + 5x - 5\), which is a polynomial of 4 terms.

Step 1: Distribute the Negative Sign

Apply the negative sign to the terms inside the second parenthesis, changing the sign of each term.

Step 2: Combine Like Terms

Identify and combine like terms from both sets of parentheses.

Step 3: Classify by Number of Terms

Count the number of terms in the simplified expression to classify it.

Key Concepts

Combining Like Terms

Combining like terms is a fundamental process in algebra which involves merging terms that have the same variables raised to the same power. For instance, in an expression like \( 3a + 4a \), both terms have the variable \(a\) and can thus be combined to \( 7a \).

When simplifying algebraic expressions like \( 7x^3 + 9x^2 - 8x + 11 \) minus \( 5x^3 - 13x - 16 \), we identify terms with the same variable and exponent. The terms \( 7x^3 \) and \( -5x^3 \) both have \(x^3\), so they can be combined by subtracting, the same goes for the \( x^2 \) and \( x \) terms, and the constant terms. By combining these like terms, we are streamlining the expression into something much more manageable and easier to work with.

Polynomial Subtraction

Polynomial subtraction involves taking away one polynomial from another. This operation is similar to normal subtraction but requires careful attention to each term. When subtracting polynomials, we address each term positionally: highest degree to lowest.

First, ensure that like terms are lined up with each other. Then, subtract coefficients of like terms while preserving the variables and their exponents. For the given problem, we subtract \( 5x^3 \) from \( 7x^3 \) and \( -13x \) from \( -8x \) and the constant term \( -16 \) from \( 11 \) respectively. This results in a new polynomial where each term is the difference of corresponding terms from the original polynomials.

Distributive Property

The distributive property, essential in algebra, allows us to distribute a factor over a sum or difference within parentheses. This means multiplying each term inside the parentheses by the factor outside. For the problem given, we see a negative sign in front of the second parentheses, which we distribute to each term inside.

Applying the distributive property changes \( -(5x^3 - 13x - 16) \) into \( -5x^3 + 13x + 16 \). It's vital to apply this property correctly, as missing the sign change can lead to incorrect results. Once distributed, we can proceed with combining like terms, taking care to accurately add or subtract coefficients.