Problem 46 - Algebra A Combined Function

Question

Factor each trinomial completely. See Examples 1 through 7. \(8 x^{2} y+34 x y-84 y\)

Step-by-Step Solution

Answer

The trinomial \(8x^2y + 34xy - 84y\) factors to \(2y(x - 2)(4x + 21)\).

Step 1: Identify Common Factor

First, let's identify any common factors in the trinomial's terms: The given trinomial is \(8x^2y + 34xy - 84y\). We notice that each term has a \(y\), which can be factored out. So, we factor \(y\) from each term, giving:\(y(8x^2 + 34x - 84)\)

Step 2: Identify the Coefficients

Next, we need to factor the quadratic expression inside the parentheses: \(8x^2 + 34x - 84\). Here, the coefficients are \(a = 8\), \(b = 34\), and \(c = -84\).

Step 3: Apply Splitting the Middle Term

We need two numbers whose product is \(a \times c = 8 \times (-84) = -672\) and sum is \(b = 34\).After testing multiple pairs, we find that the numbers are \(42\) and \(-16\), because \(42 \times (-16) = -672\) and \(42 + (-16) = 26\).Split the middle term \(34x\) using these two numbers:\(8x^2 + 42x - 16x - 84\).

Step 4: Group and Factor Pairs

Group the terms in pairs and factor each group:\((8x^2 + 42x) + (-16x - 84)\).Factor each group:- From \(8x^2 + 42x\), factor out \(2x\): \(2x(4x + 21)\).- From \(-16x - 84\), factor out \(-4\): \(-4(4x + 21)\).Now the expression is:\((2x - 4)(4x + 21)\).

Step 5: Simplify Expression

Notice that \(2x - 4\) can be further factored as \(2(x - 2)\), therefore:\((2(x - 2))(4x + 21)\).Now, write the entire expression including the common factor of \(y\):\(y \cdot 2(x - 2)(4x + 21)\).

Step 6: Present Fully Factored Form

Combine all the factors together: The completely factored form of the original trinomial is \(2y(x - 2)(4x + 21)\).

Key Concepts

Common Factor

When trying to factor trinomials, identifying a common factor is often the first step. A common factor is a number or variable that divides all terms in a polynomial. It's like finding a shared piece within each part of the expression. In our example, the trinomial is \(8x^2y + 34xy - 84y\). Each term here contains a \(y\). Therefore, \(y\) is our common factor. By factoring out \(y\) from each term, we simplify the expression to \(y(8x^2 + 34x - 84)\). This step is crucial as it reduces the complexity of the expression, making it easier to deal with in the following steps.

Quadratic Expression

A quadratic expression is a polynomial of degree two. Generally, it's formed in the structure \(ax^2 + bx + c\). In our exercise, after factoring out the common factor \(y\), the quadratic expression we are left to factor is \(8x^2 + 34x - 84\). Here, \(a = 8\), \(b = 34\), and \(c = -84\).

• The term \(ax^2\) is called the quadratic term because it has the largest exponent, which is two.
• The term \(bx\) is the linear term, involving just \(x\).
• The term \(c\) is the constant term.

Recognizing this structure is important as it helps determine the appropriate strategies you'll employ to factor the expression.

Splitting the Middle Term

Splitting the middle term is an effective method for factoring quadratic expressions when direct factoring seems challenging. It involves breaking down the middle term in a way that facilitates factoring by grouping. In our context, we work with the equation \(8x^2 + 34x - 84\). We look for two numbers that multiply to \(a \times c\) (which is \(-672\) here) and sum to \(b\) (which is \(34\) here). After testing possible pairs, \(42\) and \(-16\) fit as \(42 \times (-16) = -672\) and \(42 + (-16) = 34\). Thus, we split the middle term, \(34x\), into \(42x - 16x\), transforming the expression into \(8x^2 + 42x - 16x - 84\). This clever trick helps to set the stage for the next step in factoring.

Grouping Method

The grouping method in factoring is a technique that involves separating the expression into simpler pairs of terms and factoring each pair individually. Having transformed our quadratic expression using the splitting method, we now have \(8x^2 + 42x - 16x - 84\). We group them into pairs:

• \((8x^2 + 42x)\)
• \((-16x - 84)\)

Next, we factor each pair:- For \(8x^2 + 42x\), the common factor is \(2x\), so it becomes \(2x(4x + 21)\).- For \(-16x - 84\), the common factor is \(-4\), resulting in \(-4(4x + 21)\).Now both pairs include the same binomial factor \((4x + 21)\). Factoring this out gives \((2x - 4)(4x + 21)\). Each time we group and factor, we're simplifying the problem a bit more. In the end, by including the already factored out \(y\), the complete expression becomes \(2y(x - 2)(4x + 21)\). This reflects the power of the grouping method to turn complex expressions into manageable ones.