Problem 44 - Elementary Algebra

Question

Simplify. $$ -2 x x+6-3 x 6-x-18(x-2)(x+6)(x-6) $$

Step-by-Step Solution

Answer

Simplified expression is -18x^3 + 34x^2 + 629x - 1290.

Step 1: Distribute Terms

First, distribute and simplify the multiplication in the expression \[-2x x + 6 - 3x 6 - x - 18(x - 2)(x + 6)(x - 6)\].Expanding the first term: \[-2x x = -2x^2\]Distribute the \(-3x\) in \(-3x 6\): \[-3x 6 = -18x\].

Step 2: Combine Like Terms

Now, combine the linear and constant terms that we currently have:Starting with the expression: \[-2x^2 + 6 - 18x - x\], combine the linear terms:\[-18x - x = -19x\].The expression becomes \[-2x^2 - 19x + 6\].

Step 3: Expand Remaining Polynomial

Now expand the polynomial \[-18(x - 2)(x + 6)(x - 6)\].Start by expanding \((x + 6)(x - 6)\) using the difference of squares:\[(x + 6)(x - 6) = x^2 - 36\].

Step 4: Distribute the Multiplier

Distribute \(-18(x - 2)(x^2 - 36)\).First, expand \((x - 2)(x^2 - 36)\), which gives:\[= x^3 - 36x - 2x^2 + 72\].After distributing \(-18\),\[-18(x^3 - 36x - 2x^2 + 72) = -18x^3 + 648x + 36x^2 - 1296\].

Step 5: Combine All Terms

Combine this expression with the previously simplified terms:Combine \[-2x^2 - 19x + 6\] and \[-18x^3 + 36x^2 + 648x - 1296\].First match the terms with the same degree:1. Combine the \(x^3\) terms: only \(-18x^3\).2. Combine the \(x^2\) terms: \[36x^2 - 2x^2 = 34x^2\].3. Combine the \(x\) terms: \[648x - 19x = 629x\].4. Lastly, for integers: \[6 - 1296 = -1290\].So simplified result is \[-18x^3 + 34x^2 + 629x - 1290\].

Step 6: Present the Final Simplified Expression

The fully simplified expression is:\[-18x^3 + 34x^2 + 629x - 1290\]. Ensure you double-check each step to verify the arithmetic and distribution.

Key Concepts

Combining Like Terms

Combining like terms is an essential skill in simplifying polynomials. It involves gathering all the terms in the equation or expression that have the same variables raised to the same power. This allows us to simplify the expression into fewer, more manageable terms.

For instance, if you have the terms \(-18x - x\) in a polynomial expression, you would add these together to get \(-19x\). This is because both terms share the same variable and exponent (i.e., \(x^1\)).

• Identify terms with the same variable and exponent.
• Add or subtract the coefficients of these terms.
• Ensure the resulting expression maintains the original variables and exponents.

In the original step-by-step solution, we started with the terms\(-18x - x\), which are combined to make \(-19x\). Similarly, for constant terms and those involving higher powers, follow the same process to effectively simplify your polynomial expressions.

Distributive Property

The distributive property is a valuable tool that allows you to simplify expressions through multiplication. It states that you can distribute the multiplication across terms in parentheses. In mathematical terms, the distributive property is:\(a(b + c) = ab + ac\).

In the context of our original expression, we use the distributive property in several ways. For example, for the term \(-3x 6\), the value of \(-3x\) is distributed or multiplied by \(6\), resulting in \(-18x\).

• Multiply the number or variable outside the parentheses by each term inside the parentheses.
• Repeat distributing for all applicable terms.
• Simplify the resulting expression by combining like terms.

The distributive property simplifies the process by breaking down complicated expressions into simpler components, allowing easier combination and addition across the entire problem.

Difference of Squares

The difference of squares is a special technique for multiplying and simplifying two binomials of the form \((a + b)(a - b)\). Recognizing these products can save a lot of time in polynomial expansion.

The difference of squares formula is:\((a + b)(a - b) = a^2 - b^2\).

In the original problem, we applied this to \((x + 6)(x - 6)\), which simplifies to \(x^2 - 36\). This is because \((a + b)\) and \((a - b)\) both have the same first term (x) and second term numbers (6), leading directly to the product as outlined by the difference of squares.

• Identify pairs of binomials in the form \((a + b)(a - b)\).
• Use the formula \((a + b)(a - b) = a^2 - b^2\) to simplify the expression.

Recognizing patterns like the difference of squares can greatly reduce the complexity of polynomial expressions, saving time and minimizing possible errors.

Polynomial Expansion

Polynomial expansion is about multiplying polynomials and simplifying the result. This is a broader skill that encompasses using techniques like the distributive property or special product rules, like the difference of squares.

In our step-by-step solution, expanding \(-18(x - 2)(x + 6)(x - 6)\) requires multiple expansions and distribution to properly simplify the expression. We started by simplifying \((x + 6)(x - 6)\) with the difference of squares, yielding \(x^2 - 36\). This initial step makes further expansions manageable.

• Break down the expansion into manageable parts.
• Apply special product rules where applicable to simplify quickly.
• Use the distributive property to ensure all terms are multiplied out.

Once expanded and simplified, terms are then combined as per their degree. The key is to methodically expand each portion, simplifying where possible, resulting in a fully expanded polynomial expression with all like terms combined.