Problem 103 - College Algebra

Question

Factor completely. $$ 10 x^{2}(x+1)-7 x(x+1)-6(x+1) $$

Step-by-Step Solution

Answer

The completely factored expression is \((x+1)(5x-2)(2x+3)\)

Step 1: Identify the common factor

Upon inspection it can be observed that each term in our expression contains a factor of (x+1). This is our common factor.

Step 2: Factor out the common factor

We will now remove the common factor from each term in our expression. We are left with: \[(x+1)(10x^{2} - 7x - 6)\]. This is achieved as each term is divisible by the common factor.

Step 3: Factor the quadratic expression

Now, the task is to factorize the quadratic expression \[10x^{2} - 7x - 6\]. Look for two numbers that multiply to -60 (10*-6) and add to -7. These numbers are -12 and 5. Rewrite the middle term as the addition of terms with these coefficients and factor by grouping. The expression becomes: \[(x+1)(5x-2)(2x+3)\]

Step 4: Check the factored expression

Now, it's important to ensure everything was done properly by checking the work. This can be done by expanding the factored terms and verifying you get the original expression.

Key Concepts

Common Factor

When it comes to polynomial factoring, one of the first techniques we use is to search for a common factor. A common factor is a term that is present in each term of a polynomial. The common factor method simplifies the expression and often makes further factoring steps more manageable.

Take our example, where we have the expression: \(10x^2(x+1) - 7x(x+1) - 6(x+1)\). Each term shares \((x+1)\) making it the common factor. By factoring out \((x+1)\), we simplify the expression to \((x+1)(10x^2 - 7x - 6)\), setting up the stage for the next step in our factoring process. Remember, this initial step is pivotal for correctly and completely factoring a polynomial expression.

Factor by Grouping

When you're left with a quadratic expression, like \(10x^{2} - 7x - 6\) in our example, after factoring out the common factor, the next strategy that could be employed is factor by grouping. This involves re-arranging and grouping terms in a way that will allow us to factor further.

In practice, we look for two numbers that multiply to the product of the coefficient of \(x^2\) and the constant term. For our quadratic, we seek numbers that multiply to \(-60\) and add to \(-7\), which are \(-12\) and \(5\). These numbers allow us to break the middle term \(-7x\) and express the quadratic as \(10x^{2} + 5x - 12x - 6\). We group these into pairs, \((10x^{2} + 5x)\) and \((-12x - 6)\), which help us factor further into \((x+1)(5x-2)(2x+3)\), completing the factorization process.

Quadratic Expressions

A quadratic expression is a polynomial of degree two, typically taking the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a\) is non-zero. Quadratic expressions are foundational in algebra and occur frequently in various mathematical contexts.

In our working example, after removing the common factor from our original polynomial, we were left with the quadratic expression \(10x^{2} - 7x - 6\). Factoring such expressions requires identifying two numbers whose product equals \(ac\) (the product of the coefficient of \(x^2\) and the constant term) and whose sum equals \(b\) (the coefficient of \(x\)). These two numbers serve as a bridge to break down the quadratic expression into factors that, when multiplied, yield the original quadratic.

Polynomial Factoring

Factoring polynomials is crucial for simplifying expressions and solving equations. Polynomial factoring refers to the process of breaking down a polynomial into simpler, non-divisible polynomial factors whose product is the original polynomial. Factoring is essentially a reverse of multiplying polynomials.

To successfully factor polynomials like the one given in our exercise, one must understand and apply a variety of factoring methods including identifying a common factor, using the factor by grouping technique, and factoring quadratic expressions. Each step provides a building block towards reducing the polynomial to its simplest factor form, and the ability to move fluidly between these methods is key in solving increasingly complex polynomial equations.