Problem 54 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Binomial Theorem to expand and simplify the expression. \(-5(x+2)^{5}-2(x-1)^{2}\)

Step-by-Step Solution

Answer

The expanded and simplified expression of \(-5(x+2)^{5}-2(x-1)^{2}\) is \(-5x^{5} - 50x^{4} - 200x^{3} - 402x^{2} - 396x - 162\).

Step 1: Expand the First Term

Apply the binomial theorem to expand the expression \(-5(x+2)^{5}\). Each term in the expansion follows the pattern \(-5 \* \binom{5}{k} \* x^{5−k} \* 2^{k}\), where k is the position of the term, and the binomial coefficient \(\binom{5}{k} = \frac{5!}{k!(5−k)!}\). Expand this to obtain \(-5x^{5} - 50x^{4} - 200x^{3} - 400x^{2} - 400x -160\).

Step 2: Expand the Second Term

Follow the same process for \(-2(x-1)^{2}\), resulting in the expansion \(-2x^{2} + 4x - 2\).

Step 3: Combine Terms

Sum the expanded expressions from the previous steps: \(-5x^{5} - 50x^{4} - 200x^{3} - 400x^{2} - 400x - 160 - 2x^{2} + 4x - 2\)

Step 4: Simplify Expression

Combine like terms to simplify the expression, and yield the final simplified expression: \(-5x^{5} - 50x^{4} - 200x^{3} - 402x^{2} - 396x - 162\)

Key Concepts

Expanding Expressions

Expanding expressions using the Binomial Theorem can feel like opening a magical box full of terms. This process allows us to transform expressions like \((x+2)^5\) into a long polynomial. The Binomial Theorem helps to break out each term systematically. To expand, identify the structure as \((a+b)^n\), where \(a\) and \(b\) represent distinct terms of the binomial, and \(n\) is the power to which they are raised. The theorem generates terms using the format \( \binom{n}{k} a^{n-k} b^k \), where \(k\) begins at 0 and ends at \(n\). Each term gets multiplied by a binomial coefficient and respective powers of \(a\) and \(b\). Studying each part can help:

• Increase efficiency in expanding binomials.
• Identify patterns in the resulting polynomial.
• Improve accuracy by systematically applying the theorem.

In our problem, this process led us to expand both \(-5(x+2)^5\) and \(-2(x-1)^2\) into separate polynomial forms.

Polynomial Coefficients

Polynomial coefficients are critical elements when expanding expressions. They dictate the "weight" of each term. In binomial expansions, these coefficients come from binomial coefficients \(\binom{n}{k}\), calculated using factorials.Remember, factorials represent the product of all positive integers up to a given number. This provides a way to determine the combinatorial aspect of the coefficients. In our case, each term of \(-5(x+2)^5\) gets multiplied by such binomial coefficients. For every position \(k\), \(\binom{5}{k}\) is calculated to determine how many times that particular term configuration appears. This is essential for:

• Precise calculation of the polynomial's components.
• Avoiding mistakes by systematically using combinatorial math.
• Increasing algebraic manipulation skills.

Grasping polynomial coefficients helps in crafting a correct expanded expression, guiding students towards mastering polynomials.

Combining Like Terms

Combining like terms is the final polishing step in simplifying expanded polynomials. Once you've whipped out all terms with the help of expansions, the next goal is to streamline them. Like terms are terms with the same variable raised to the same power. For example, terms like \(-400x^2\) and \(-2x^2\) can be combined.Combining involves:

• Adding or subtracting the coefficients of like terms.
• Ensuring all terms are reduced to their simplest form.
• Checking and rechecking accuracy in combined values.

The problem given then simplifies to \(-5x^5 - 50x^4 - 200x^3 - 402x^2 - 396x - 162\). By learning how to combine like terms, you ensure your final expression is tidy and ready for any further mathematical action like graphing or factoring.