Problem 62 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Factor the polynomial. $$ 10 x^2+31 x-14 $$

Step-by-Step Solution

Answer

The polynomial \(10x^2+31x-14\) factors as \((5x-2)(2x+7)\)

Step 1: Identify the coefficients

Identify the coefficients a, b, c from the quadratic equation \(10x^2 + 31x - 14\). Here, 'a' is 10, 'b' is 31 and 'c' is -14.

Step 2: Find two numbers that add to 'b' and multiply to 'ac'

Find two numbers that add up to 'b' (which is 31 in this case) and multiplies to 'ac' (which is -140 in this case, as 'a' is 10 and 'c' is -14). After some calculations, we find the numbers 35 and -4 which satisfy the condition \(35 × -4 = -140\) and \(35 - 4 = 31\).

Step 3: Split the middle term

Next step is to split the middle term 'bx' using the two numbers we found. So re-write the equation as \(10x^2 + 35x -4x - 14\).

Step 4: Factor by grouping

Now, factor by grouping: We first group \(10x^2+35x\) and \(-4x-14\) separately. Then within each group, factor out a greatest common factor (GCF). The GCF of \(10x^2+35x\) is \(5x\) and the GCF of \(-4x-14\) is -2. After factoring, we have \(5x(2x+7) -2(2x+7)\).

Step 5: Combine like terms

Finally, notice that the factors \((2x+7)\) are the same, so combine the terms with the same factor to get the final factored form: \((5x-2)(2x+7)\) as the factored form of the given polynomial.

Key Concepts

Understanding Quadratic Equations

Quadratic equations are polynomials of degree two, which means they involve a squared term as the highest power of the variable, typically denoted by \(x^2\). These types of equations generally take the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants with \(a eq 0\). The value of \(a\) cannot be zero because, without the \(x^2\) term, it would no longer be a quadratic equation.

Quadratic equations are fundamental in algebra and appear in numerous applications across different fields. They can have different types of solutions based on the discriminant, which is the part of the quadratic formula under the square root, \(b^2 - 4ac\). Understanding how to manipulate these equations through techniques like factoring is crucial for solving them effectively.

Coefficients in Quadratic Equations

Coefficients are the numerical factors that multiply the variables in a polynomial. In the context of a quadratic equation \(ax^2 + bx + c\), these are the numbers \(a\), \(b\), and \(c\).

Each of these coefficients serves a specific role:

• \(a\): Dictates the opening direction of the parabola—whether it opens upwards or downwards—and also influences its steepness.
• \(b\): Affects the position of the vertex horizontally and has an impact on the parabola's axis of symmetry.
• \(c\): Often represents the y-intercept of the graph, giving the point where the parabola crosses the y-axis.

In any factoring or solving technique, identifying these coefficients correctly is a crucial first step. In our exercise, \(a = 10\), \(b = 31\), and \(c = -14\), which guide the factorization process.

Factoring by Grouping

Factoring by grouping is a method used to simplify polynomials, especially useful for quadratics that don't easily factor into simple binomials. It involves rearranging and dividing the polynomial into groups that share a common factor, which can then be factored out to simplify the expression further.

Here's how it works in the context of our exercise:
1. **Identify groups**: Split the middle term to form two groups: \(10x^2 + 35x\) and \(-4x - 14\).
2. **Factor each group**: Find the greatest common factor for each group: - For \(10x^2 + 35x\), the GCF is \(5x\). - For \(-4x - 14\), the GCF is \(-2\). 3. **Factor out the GCF**: Pull out the common factor from each group: - \(5x(2x + 7)\) for the first group. - \(-2(2x + 7)\) for the second group.
After grouping and factoring, notice the same binomial \((2x + 7)\) occurs in both terms, allowing a single act of combination. This results in the factored form \((5x - 2)(2x + 7)\). Factoring by grouping is a powerful technique, especially when the standard quadratic "ax^2 + bx + c" form doesn't offer obvious factors.

Greatest Common Factor (GCF)

The greatest common factor, or GCF, is the largest factor shared by two or more numbers or terms. Finding the GCF is essential when factoring expressions as it allows you to simplify the polynomial and make further factorization easier.

In the step-by-step solution, when identifying and then pulling out the GCF from the groups \(10x^2 + 35x\) and \(-4x - 14\), the process looked like this:

• For \(10x^2 + 35x\): The common factor is \(5x\).
• For \(-4x - 14\): The common factor is \(-2\).

The GCF simplifies each part by reducing unnecessary complexity, making the final factorization more straightforward. By factoring out these common elements, what remains inside the parentheses can be made the same (\((2x + 7)\)), allowing the polynomial to be expressed in a simpler, factored form. Recognizing and using the GCF in factoring is a basic yet powerful tool in algebra that strengthens the ability to solve polynomial equations efficiently.