Algebra Practice Notes: Polynomials and Rational Expressions

Polynomial Multiplication Practices

Question 1: Product of Binomial and Trinomials

Problem Statement

The question asks to find the product of two polynomials: (a-3)(2a^2 - a + 1).

Key Concept: Distributive Property

To multiply a binomial by a trinomial, we use the extended distributive property. Each term in the first polynomial (the binomial) must be multiplied by each term in the second polynomial (the trinomial). After distribution, combine any like terms to simplify the expression.

Solution Steps

  1. Distribute the first term of the binomial (a) to each term in the trinomial (2a^2, -a, 1).
  2. Distribute the second term of the binomial (-3) to each term in the trinomial (2a^2, -a, 1).
  3. Write out all the resulting terms.
  4. Identify and combine like terms (terms with the same variable and exponent).

Detailed Calculation

Given the expression: (a-3)(2a^2 - a + 1)

  1. Distribute a:
    a(2a^2) = 2a^{2+1} = 2a^3
    a(-a) = -a^{1+1} = -a^2
    a(1) = a
    So, the first part is: 2a^3 - a^2 + a

  2. Distribute -3:
    -3(2a^2) = -6a^2
    -3(-a) = +3a
    -3(1) = -3
    So, the second part is: -6a^2 + 3a - 3

  3. Combine the results from steps 1 and 2:
    (2a^3 - a^2 + a) + (-6a^2 + 3a - 3)
    = 2a^3 - a^2 + a - 6a^2 + 3a - 3

  4. Combine like terms:
    Combine a^2 terms: -a^2 - 6a^2 = -7a^2
    Combine a terms: a + 3a = 4a
    The constant term remains: -3
    The a^3 term remains: 2a^3

    Therefore, the simplified product is: 2a^3 - 7a^2 + 4a - 3

Result and Option Match

The calculated product is 2a^3 - 7a^2 + 4a - 3. Among the provided options (assuming they are interpreted as standard mathematical expressions despite formatting eccentricities):

  • Option (g) 2a37a2 + 4a-3 most closely matches our result, interpreting 2a37a2 as 2a^3 - 7a^2 and + 4a-3 as + 4a - 3. Thus, option (g) is equivalent to 2a^3 - 7a^2 + 4a - 3.

Simplifying Rational Expressions

Question 2: Sum of Fractions with Denominators Involving Radicals/Constants

Problem Statement

The question asks to simplify the expression: (1/(4+3)) + (1/(4-3))

Ambiguity in Interpretation and Key Concepts

This question presents an ambiguity due to the notation of '3' in the denominators. It is common in algebra problems for '3' in such a context to implicitly mean " ext{the square root of } 3" (i.e., " ext{sqrt}(3)" or "ackslash ext{sqrt}{3}" using LaTeX). However, based on the literal transcript, it appears as a simple integer '3'. Both interpretations lead to different results.

Interpretation 1: Denominators with Radicals (Common Algebraic Problem Type)

If the '3' in the denominators actually represents " ext{the square root of } 3" (ackslash ext{sqrt}{3}), this is a classic problem involving conjugates to rationalize denominators.

  • Key Concept: To add fractions, a common denominator is required. When denominators involve binomials with radicals (e.g., a+backslash ext{sqrt}{c}), the common denominator is often formed by multiplying the denominators, using the property of conjugates: (x+y)(x-y) = x^2 - y^2. This eliminates the radical from the denominator.
Detailed Calculation (Assuming ackslash ext{sqrt}{3})

Expression: (1/(4+ackslash ext{sqrt}{3})) + (1/(4-ackslash ext{sqrt}{3}))

  1. Find the common denominator. This is the product of the two existing denominators: (4+ackslash ext{sqrt}{3})(4-ackslash ext{sqrt}{3}).
    Using the difference of squares formula, x^2 - y^2 where x=4 and y=ackslash ext{sqrt}{3}.
    (4+ackslash ext{sqrt}{3})(4-ackslash ext{sqrt}{3}) = 4^2 - (ackslash ext{sqrt}{3})^2 = 16 - 3 = 13
    So, the common denominator is 13.

  2. Rewrite each fraction with the common denominator:
    For the first fraction, multiply numerator and denominator by (4-ackslash ext{sqrt}{3}):
    (1/(4+ackslash ext{sqrt}{3})) * ((4-ackslash ext{sqrt}{3}) / (4-ackslash ext{sqrt}{3})) = (4-ackslash ext{sqrt}{3}) / ((4+ackslash ext{sqrt}{3})(4-ackslash ext{sqrt}{3})) = (4-ackslash ext{sqrt}{3}) / 13

    For the second fraction, multiply numerator and denominator by (4+ackslash ext{sqrt}{3}):
    (1/(4-ackslash ext{sqrt}{3})) * ((4+ackslash ext{sqrt}{3}) / (4+ackslash ext{sqrt}{3})) = (4+ackslash ext{sqrt}{3}) / ((4-ackslash ext{sqrt}{3})(4+ackslash ext{sqrt}{3})) = (4+ackslash ext{sqrt}{3}) / 13

  3. Add the two modified fractions:
    ( (4-ackslash ext{sqrt}{3}) / 13 ) + ( (4+ackslash ext{sqrt}{3}) / 13 ) = (4-ackslash ext{sqrt}{3} + 4+ackslash ext{sqrt}{3}) / 13

  4. Simplify the numerator:
    4+4 - ackslash ext{sqrt}{3} + ackslash ext{sqrt}{3} = 8

    Therefore, the simplified expression is: 8/13

Interpretation 2: Denominators as Simple Integers (Literal Reading)

If the '3' in the denominators is simply the integer 3, then the problem is a straightforward addition of numerical fractions.

  • Key Concept: To add fractions, a common denominator is required. Calculate the values of the denominators first, then find their least common multiple or product to use as the common denominator.
Detailed Calculation (Assuming 3 as an Integer)

Expression: (1/(4+3)) + (1/(4-3))

  1. Evaluate the denominators:
    4+3 = 7
    4-3 = 1

  2. The expression becomes: (1/7) + (1/1)

  3. Find a common denominator, which is 7. Rewrite the second fraction:
    1/1 = 7/7

  4. Add the fractions:
    1/7 + 7/7 = (1+7) / 7 = 8/7

    Therefore, the simplified expression is: 8/7

Analysis of Options and Discrepancy

The provided options are extremely ambiguous and poorly formatted (e.g., 12-9, 21-12). Interpreting them as fractions (Numerator-Denominator representing Numerator/Denominator) or simple numeric differences does not yield a clear match for either of our calculated results (8/13 or 8/7).

Given the strong likelihood of a missing radical symbol common in transcribed math problems, the solution 8/13 (under Interpretation 1) is generally what such a problem is designed to test. However, without a correctly formatted set of options, a definitive answer choice cannot be confirmed from the transcript. The steps provided cover the most probable approaches to solving this type of expression.