Problem 32 - Algebra 1

Question

Simplify the expression. $$\frac{6 x^{2}+7 x-33}{x+4} \div(6 x-11)$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{2x+3}{x+4}\)

Step 1: Reciprocal of the second term

Turn the division into multiplication by taking the reciprocal of the second term. It turns the expression into \(\frac{6 x^{2}+7 x-33}{x+4} * \frac{1}{6x-11}\)

Step 2: Factorize the numerator

Factorization of the quadratic \(6 x^{2}+7 x-33\) can be done. The factors are \(3x-11\) and \(2x+3\). The expression becomes \(\frac{(3x-11)(2x+3)}{x+4} * \frac{1}{6x-11}\)

Step 3: Cancel common terms

We can now cancel the common term \(3x-11\) from the numerator and the denominator. The expression simplifies to \(\frac{2x+3}{x+4}\)

Key Concepts

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into simpler, multiplied factors. This process is essential for simplifying algebraic expressions and solving equations. Let's consider a polynomial like \(6x^2 + 7x - 33\). To factorize this polynomial, we must find two binomials whose product equals the original polynomial.

• First, identify a pair of numbers that multiply to the product of the quadratic term's coefficient and the constant term (in this case, \(6 \times -33 = -198\)) and add up to the linear term's coefficient (which is 7).
• The numbers 18 and -11 work, as they multiply to -198 and add up to 7.
• Rewrite the middle term using these numbers: \(6x^2 + 18x - 11x - 33\).
• Then, factor by grouping: \((6x^2 + 18x) + (-11x - 33)\) becomes \(6x(x + 3) - 11(x + 3)\).
• Finally, factor out the common terms: \((6x - 11)(x + 3)\).

Each step simplifies the polynomial, making further operations easier.

Division of Algebraic Expressions

The division of algebraic expressions involves turning division into multiplication by using the reciprocal of the divisor. For the given expression, replace the division operation:

• Change \(\frac{6x^2 + 7x - 33}{x+4} \div (6x-11)\) into \(\frac{6x^2 + 7x - 33}{x+4} \times \frac{1}{6x-11}\).
• This adjustment allows for simpler computations and aligns with the multiplication property.
• Now, combine the two fractions into one by multiplying their numerators and denominators: \(\frac{(6x^2 + 7x - 33) \cdot 1}{(x+4)(6x-11)}\).

Such transformation takes advantage of factorization, ensuring clarity in solving problems and enabling cancellations that bring the expression to its simplest form.

Rational Expressions

Rational expressions are fractions involving polynomials in their numerators and denominators. Learning how to simplify these expressions is crucial for advancing in algebra. Let's take an expression like \(\frac{6x^2 + 7x - 33}{x+4}\):

• After factorizing the numerator to \((3x-11)(2x+3)\), rational expressions allow us to cancel out common terms between the numerator and denominator.
• In our example, \(3x-11\) was a factor in both the original numerator's factorization and the divisor \(6x-11\), thus it can be canceled.
• Post-cancellation, the simplified expression is \(\frac{2x+3}{x+4}\).

By simplifying rational expressions, we reduce complexity in calculations and improve the readability of algebraic functions.