Problem 83 - Beginning and Intermediate Algebra

Question

Factor completely by first taking out a negative common factor. $$-16 y^{2}-34 y+15$$

Step-by-Step Solution

Answer

The short answer for factoring the given expression completely is: \[-(7a^2 + 4a + 3) = (7a + 1)(a - 3).\]

Step 1: Identify the common factor

First, let's identify the negative common factor. In this case, the common factor is -1, as all of the terms have a negative sign.

Step 2: Factor out the common factor

Now, let's factor out the negative common factor (-1) from each term. To do this, divide each term by -1. \[ -1(-7a^2 + 4a + 3) \] Upon factoring -1 from each term, we get: \[ 1(7a^2 - 4a - 3) \] Since multiplying by 1 doesn't change the expression, we can just rewrite this as: \[ 7a^2 - 4a - 3 \]

Step 3: Factor the quadratic expression

Now, let's attempt to factor the remaining quadratic expression completely. To factor a quadratic expression in the form \(ax^2 + bx + c\), we look for two numbers that multiply to 'c' (the constant term) and add up to 'b' (the coefficient of the linear term). In our case, a = 7, b = -4, and c = -3. We look for two numbers that multiply to -3 and add up to -4. The numbers -3 and 1 fit these conditions, and we can thus factor the quadratic expression as follows: \[ 7a^2 - 4a - 3 = (7a + 1)(a - 3) \] So, the factored form of the given expression is: \[ -1(7a^2 + 4a + 3) = (7a + 1)(a - 3) \]

Key Concepts

Negative Common Factor

When dealing with polynomials, one effective strategy for simplification is factoring out a common factor. In some cases, this common factor is negative. Identifying and factoring out a negative common factor can simplify the expression and make further operations easier. This technique involves:

• Checking each term of the polynomial for a negative sign.
• Recognizing negative coefficients as an indicator of a negative common factor.

Once identified, you can factor out this negative factor (often \(-1\) for polynomials) by dividing each term by \(-1\). This step can change the signs of all the terms, simplifying further operations. For example, if you have \(-16y^2 - 34y + 15\), factoring out \(-1\) changes the expression to \(16y^2 + 34y - 15\). This transformation helps eliminate negative signs, making it easier to visualize and perform subsequent factoring.

Quadratic Expression

A quadratic expression takes the general form \(ax^2 + bx + c\). These expressions are mathematical statements that might look complicated at first but can usually be factored into simpler binomial expressions. Understanding quadratics involves recognizing the roles of:

• \(a\) - the coefficient of the squared term
• \(b\) - the coefficient of the linear term
• \(c\) - the constant term

Each part plays a vital role in factoring and can help set up for more straightforward solutions.
To factor a quadratic expression, being able to find two numbers that multiply to \(c\) and add up to \(b\) is crucial. For instance, given \(7a^2 - 4a - 3\), the aim is to break it down, so it leads to a product of binomials, such as \((7a + 1)(a - 3)\). This factorization process is a foundational tool in algebra, frequently utilized to simplify expressions and solve quadratic equations.

Factoring Completely

Factoring completely is a method used to simplify polynomials entirely, leaving them in their simplest multipliable parts. This means not only identifying a common factor but also breaking down the quadratic or polynomial expressions until no further factoring is possible. While factoring completely you'll:

• Identify and extract any common factors.
• Use methods like grouping, special products patterns, or trial and error to factor the quadratic expression.
• Ensure that the expression is expressed as the product of irreducible factors.

In the example of \(-16y^2 - 34y + 15\), after factoring out the \(-1\), you're left with \(7a^2 - 4a - 3\). Continuing with factoring completely involves identifying the binomials that satisfy the given equation, in this case, \((7a + 1)(a - 3)\), to achieve the simplest form. Mastering this skill involves practice and familiarity with factoring patterns, yet successfully doing so can reveal solutions hidden in complex expressions.