Problem 48 - Elementary Algebra

Question

Factor \(8 x^{2}-18 x-5\).

Step-by-Step Solution

Answer

Answer: The factored form of \(8x^2 - 18x - 5\) is \((2x - 5)(4x + 1)\).

Step 1: Calculate "ac" and find two numbers that multiply to "ac" and add to "b"

In our quadratic expression, \(a = 8\) , \(b = -18\), and \(c = -5\). Let's calculate \(ac\): \(ac = (8)(-5) = -40\) We need to find two numbers that multiply to -40 and add up to -18. The two numbers that fit this criteria are -20 and 2. (-20)(2) = -40 and (-20) + 2 = -18

Step 2: Rewrite the expression using the numbers found in Step 1

Now we can rewrite the middle term, \(-18x\), as a sum of two terms using the two numbers we found in step 1: \(8x^2 - 18x - 5 = 8x^2 - 20x + 2x - 5\)

Step 3: Factor by grouping

Next, we factor by grouping the terms: \(8x^2 - 20x + 2x - 5 = (8x^2 - 20x) + (2x - 5)\) Now, we factor out the greatest common factor from each group: \((8x^2 - 20x) = 4x(2x - 5)\) and \((2x - 5) = 1(2x - 5)\) So our expression now looks like this: \(4x(2x - 5) + 1(2x - 5)\) Both terms have the common factor of \((2x - 5)\), so we can factor that out to obtain our final answer: \((2x - 5)(4x + 1)\) So, the factored form of \(8x^2 - 18x - 5\) is \((2x - 5)(4x + 1)\).

Key Concepts

Quadratic Expressions

In mathematics, quadratic expressions are polynomials of degree 2. These expressions are generally written in the standard form as \[ ax^2 + bx + c, \]where \(a, b,\) and \(c\) are constants, and \(a eq 0\). The term \(ax^2\) is the quadratic term, \(bx\) is the linear term, and \(c\) is the constant term.

Quadratic expressions can often be factored, which is a process of breaking them down into simpler components that, when multiplied together, give back the original expression. The roots of the quadratic expression are the values of \(x\) which make the expression equal to zero. Understanding how to manipulate these expressions is crucial in solving quadratic equations, graphing parabolas, and modeling real-life scenarios where quadratic relationships occur.

Factoring by Grouping

Factoring by grouping is a powerful method used to factor polynomials, especially when dealing with quadratic expressions with a leading coefficient other than 1. This technique involves four main steps:

• Step 1: Multiply \(a\) (the coefficient of \(x^2\)) and \(c\) (the constant term).
• Step 2: Identify two numbers that multiply to \(ac\) and add up to \(b\) (the coefficient of \(x\)).
• Step 3: Split the middle term using the two numbers found.
• Step 4: Group the terms and factor out the greatest common factor from each group, then factor the common binomial completely.

This method is especially useful when traditional factoring techniques do not easily apply. By organizing the terms into groups, the expression remains manageable and leads more quickly to a solution.

Greatest Common Factor

The Greatest Common Factor (GCF) is a fundamental concept in factorization, both for numbers and algebraic expressions. When factoring, the GCF is the largest factor that is common to all terms in a polynomial. Finding the GCF can simplify the initial step of the factoring process.

To identify the GCF:

• Identify the factors of each term.
• Choose the highest factor common across all terms.
• Factor out the GCF from each group of terms.

For example, in the polynomial \(8x^2 - 20x\), both terms have \(4x\) as a common factor, which simplifies the expression dramatically. Recognizing and using the GCF effectively simplifies problems and is essential in factoring quadratics and more complex polynomials.

Polynomials

Polynomials are expressions that include variables raised to whole number powers and coefficients. A polynomial of one variable \(x\) is typically expressed as:\[a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0,\]where the exponents are non-negative integers, and \(a_n\) is not zero. The degree of the polynomial is the highest exponent of the variable.

For quadratic polynomials, this degree is 2. Recognizing the degree helps determine how to approach factoring the polynomial. Polynomials are foundational in algebra due to their varied applications, from solving equations to modeling curves in geometric spaces. They form the building blocks for more advanced mathematics and appear frequently in real-world applications, highlighting their importance in both theoretical and applied mathematics.