Problem 14 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a polynomial. $$ \left(r^{2}-8 r-2\right)\left(-r^{2}+3 r-1\right) $$

Step-by-Step Solution

Answer

The polynomial is \(-r^4 + 11r^3 - 23r^2 + 2r + 2\).

Step 1: Distribute the First Term

Start by distributing each term of the first binomial to the second binomial. Begin with \( r^2 \). Multiply \( r^2 \) by each term in the second binomial: \[r^2 (-r^2) + r^2 (3r) + r^2 (-1) = -r^4 + 3r^3 - r^2\]

Step 2: Distribute the Second Term

Next, distribute \(-8r\) to each term in the second binomial: \[-8r (-r^2) + (-8r) (3r) + (-8r) (-1) = 8r^3 - 24r^2 + 8r\]

Step 3: Distribute the Third Term

Finally, distribute \(-2\) to each term in the second binomial: \[-2 (-r^2) + (-2) (3r) + (-2) (-1) = 2r^2 - 6r + 2\]

Step 4: Combine Like Terms

Combine all the resulting terms from each distribution to form a single polynomial by combining like terms: - Add the \( r^4 \) terms: \( -r^4 \)- Add the \( r^3 \) terms: \( 3r^3 + 8r^3 = 11r^3 \)- Add the \( r^2 \) terms: \( -r^2 - 24r^2 + 2r^2 = -23r^2 \) - Add the \( r \) terms: \( 8r - 6r = 2r \) - Constant term = \( 2 \).The final polynomial is: \[-r^4 + 11r^3 - 23r^2 + 2r + 2\]

Key Concepts

Binomial Distribution

Binomial Distribution is a method used to expand expressions that multiply two binomials together. A binomial is an algebraic expression containing two terms. In the given exercise, you are multiplying two binomials, \( (r^2 - 8r - 2) \) and \( (-r^2 + 3r - 1) \). To distribute a binomial, follow these steps:

• Select the first term from the first binomial and multiply it by each term from the second binomial.
• Continue by selecting the second term from the first binomial and again multiply it by each term from the second binomial.
• Repeat this process for any additional terms in the binomial.

This process ensures that every term in the first binomial is multiplied with every term in the second binomial, covering all possible combinations without missing any terms. The aim is to simplify and eventually write the expression as a single polynomial.

Combining Like Terms

Combining Like Terms is the process of reducing an expression by merging terms that have identical variables and exponents. After you have distributed terms from each binomial in the exercise, like terms from the results need to be combined to simplify the expression further.

• The terms with the same variable and power should be summed together.
• Ensure you keep track of the mathematical signs in front of each term.

For example, in the solution, after distributing and getting terms like \( 3r^3 \) and \( 8r^3 \), these terms are combined to give \( 11r^3 \). This process creates a cleaner expression by minimizing the number of individual terms you need to manage.Combining like terms is crucial in polynomial arithmetic because it simplifies the expansion to its shortest form, making it easier to interpret and solve.

Polynomial Expansion

Polynomial Expansion involves expressing a multiplication of polynomials in a standard polynomial format. This exercise showcased expanding two binomials into a polynomial using the distribution method. The final expanded form is a polynomial that sums all products obtained from the distribution process to offer a more comprehensive and simpler expression.

• You start with two or more polynomial expressions.
• Apply the distributive property methodically to expand these expressions.
• Simplify by combining like terms.

The outcome is a polynomial typically written in descending order based on the power of terms. In our example, you begin with the highest power, \(-r^4\), and descend to the constant term, \(2\). Successfully expanding and simplifying a polynomial enhances understanding of how functions behave and provides a groundwork for further mathematical analysis.