Problem 88 - Intermediate Algebra

Question

a. \(\frac{9 a^{3}+3 a^{2}+4 a+4}{3 a}\) b. \(\frac{9 a^{3}+3 a^{2}+4 a+4}{3 a+2}\)

Step-by-Step Solution

Answer

a. The simplified form is \(3a^2 + 4\). b. The simplified form is \(3a^2 - a + 2\).

Step 1: Simplifying Expression a

For part (a), the expression is \( \frac{9a^3 + 3a^2 + 4a + 4}{3a} \). Start by factoring the numerator, \( 9a^3 + 3a^2 + 4a + 4 \). Group as \( (9a^3 + 3a^2) + (4a + 4) \) and factor each group: \( 3a^2(3a + 1) + 4(a + 1) \). Notice that \( a + 1 \) is a common factor, giving \( (3a^2 + 4)(3a + 1) \). The expression simplifies by canceling out \( 3a \) with part of the numerator: \( \frac{3a^2 + 4}{1} = 3a^2 + 4 \).

Step 2: Simplifying Expression b

For part (b), consider \( \frac{9a^3 + 3a^2 + 4a + 4}{3a + 2} \). A different approach is needed. Use polynomial division to divide \( 9a^3 + 3a^2 + 4a + 4 \) by \( 3a + 2 \). Determine the first term by dividing \( 9a^3 \) by \( 3a \), which gives \( 3a^2 \). Multiply \( 3a^2 \) by \( 3a + 2 \) resulting in \( 9a^3 + 6a^2 \). Subtract this from the original polynomial, resulting in \(-3a^2 + 4a + 4 \).

Step 3: Continuing Polynomial Division

Continue the division process: divide \(-3a^2\) by \(3a\) to get \(-a\). Multiply resulting \(-a\) by \(3a + 2\), giving \(-3a^2 - 2a\). Subtract \(-3a^2 - 2a\) from \(-3a^2 + 4a + 4\), resulting in \(6a + 4\).

Step 4: Finishing Polynomial Division

For the last step, divide \(6a\) by \(3a\), resulting in \(2\). Multiply \(2\) by \(3a + 2\), giving \(6a + 4\). Subtract to reach 0. Hence, the expression simplifies to \(3a^2 - a + 2\).

Step 5: Final Simplified Expressions

The fully simplified result for part a is \(3a^2 + 4\). For part b, it is \(3a^2 - a + 2\), demonstrating complete division with no remainder.

Key Concepts

Factoring Polynomials

Factoring polynomials involves breaking down a complex polynomial expression into simpler components or factors that, when multiplied, give the original polynomial back. This is a crucial step in simplifying many algebraic expressions and solving equations.
To effectively factor a polynomial, start by looking for common factors in all terms. For example, in the polynomial \(9a^3 + 3a^2 + 4a + 4\), notice how it can be grouped into \((9a^3 + 3a^2) + (4a + 4)\). Each group can then be factored individually:

• Factor out \(3a^2\) from \(9a^3 + 3a^2\) to get \(3a^2(3a + 1)\).
• Factor out \(4\) from \(4a + 4\) to get \(4(a + 1)\).

Recognize any identical binomial factors, like \(a + 1\), and factor these further: \((3a^2 + 4)(3a + 1)\). By mastering these strategies, students can simplify many polynomial expressions more efficiently.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing a fraction-like algebraic expression to its simplest form. These expressions often appear in algebra and require careful manipulation for simplification.
The process of simplifying a rational expression such as \(\frac{9a^3 + 3a^2 + 4a + 4}{3a}\) begins by factoring both the numerator and the denominator. Once they are fully factored, any common factors can be canceled out. For example, after factoring the numerator into \((3a^2 + 4)(3a + 1)\), the denominator \(3a\) can cancel out part of the numerator to yield a simpler result of \(3a^2 + 4\).
Key steps in simplifying rational expressions include:

• Factor both the numerator and denominator.
• Identify and cancel common factors.
• Rewrite the expression in simplified form.

These actions help in creating a more manageable expression that retains the same value as the original.

Algebraic Long Division

Algebraic long division is a method used to divide polynomials, analogous to traditional long division with numbers. It's particularly useful when simplifying expressions that do not factor neatly.
Consider the example \(\frac{9a^3 + 3a^2 + 4a + 4}{3a + 2}\). Start by dividing the leading term of the numerator, \(9a^3\), by the leading term of the denominator, \(3a\), resulting in \(3a^2\). Multiply \(3a^2\) by \(3a + 2\), subtract from the original polynomial, and proceed similarly with each new smaller polynomial form obtained:

• Step 1: \(-3a^2 + 4a + 4\)
• Step 2: Divide \(-3a^2\) by \(3a\) to get \(-a\).
• Step 3: Multiply and subtract to get \(6a + 4\).
• Step 4: Finally, divide \(6a\) by \(3a\) to obtain \(2\).

When done correctly, you'll reach a remainder of 0, with the final quotient being \(3a^2 - a + 2\). This entire process provides a precise and systematic way to handle polynomial division tasks.