Problem 67 - Elementary Algebra

Question

For the following problems, perform the indicated operations. $$ (x-7)^{4} \div \frac{(x-7)^{3}}{x+1} $$

Step-by-Step Solution

Answer

Question: Simplify the following expression: $(x-7)^4 \div \frac{(x-7)^3}{x+1}$ Answer: $x^2 - 6x - 7$

Step 1: Change division to multiplication

To get rid of the division, we can multiply by the reciprocal of the fraction. This means we need to flip the fraction and change the division operation to multiplication. So the given expression becomes: $$(x-7)^4 \cdot \frac{x+1}{(x-7)^3}$$

Step 2: Simplify the expression using the power properties

Now, we can use the power properties to simplify this expression further. We will use the property \(a^m \cdot a^n = a^{m+n}\) to combine the powers of \((x-7)\): $$(x-7)^4 \cdot \frac{(x+1)}{(x-7)^3} = (x-7)^{4-3} \cdot (x+1) = (x-7)^1 \cdot (x+1)$$ We can then simplify \((x-7)^1\) to just \((x-7)\): $$(x+1)(x-7)$$

Step 3: Expand the expression

Now, we have simplified our expression into a product of two binomials. We can expand this expression by using the distributive property: $$(x+1)(x-7) = x(x-7) + 1(x-7) = x^2 -7x + x -7$$

Step 4: Combine like terms

Finally, we will combine the like terms (the terms with the same variables and powers) to get the final simplified expression: $$x^2 -7x + x -7 = x^2 - 6x - 7$$ Therefore, the simplified expression is: $$ (x-7)^4 \div \frac{(x-7)^3}{x+1} = x^2 - 6x - 7 $$

Key Concepts

Polynomial Division

Polynomial division is a process similar to dividing numbers. It involves dividing one polynomial by another. In algebra, polynomials are expressions consisting of variables, coefficients, and operations like addition, subtraction, and multiplication. Division can also include fractions where the numerator and denominator are polynomials. With polynomial division, the goal is often to simplify the expression. A common approach to handle division is to multiply by the reciprocal, just as we do with numbers. For example, consider dividing by \( \frac{a}{b} \), which is equivalent to multiplying by \( \frac{b}{a} \).

• This transformation allows us to switch the division into multiplication.
• It simplifies the handling of expressions, as seen in the exercise of dividing \((x-7)^4\) by \(\frac{(x-7)^3}{x+1}\).

By multiplying \((x-7)^4\) by \(\frac{x+1}{(x-7)^3}\), we convert the division problem into a multiplication problem, simplifying the polynomial manipulation process.

Exponents and Powers

Exponents and powers are fundamental algebraic concepts used to express repeated multiplication. When you see something like \( a^n \), it means you multiply \( a \) by itself \( n \) times.

• They follow specific rules that are essential for simplifying expressions.
• One key rule is that when multiplying like bases, you add the exponents: \( a^m \times a^n = a^{m+n} \).

This rule was applied in the original exercise to simplify \((x-7)^4 \times \frac{1}{(x-7)^3}\).By applying the properties of exponents:

• The expression becomes \((x-7)^{4-3} = (x-7)^1\).

This simplifies to just \((x-7)\), making the expression easier to expand further. Mastering the laws of exponents helps streamline many algebraic operations.

Distributive Property

The distributive property is a critical property in algebra for simplifying expressions involving multiplication over addition or subtraction. It is expressed as \( a(b+c) = ab + ac \), indicating that you multiply every term within the parentheses by the term outside.In the exercise, using the distributive property helped us to expand the polynomials.

• The expression \((x+1)(x-7)\) requires us to distribute each term of one binomial to every term in the other binomial.
• This becomes \(x(x-7) + 1(x-7)\), which upon simplification results in \(x^2 - 7x + x - 7\).

Finally, combining like terms is essential after using the distributive property. In this example:

• Combining \(-7x + x\) results in \(-6x\).

Thus, the expression simplifies to \(x^2 - 6x - 7\), showing the effectiveness of the distributive property in managing and simplifying polynomial expressions.