Question

Find the real solutions of each equation by factoring. $$ x^{3}+6 x^{2}-7 x=0 $$

Short answer

The real solutions are \(x = 0\), \(x = -7\), and \(x = 1\).

Step-by-step solution

Factor out the Greatest Common Factor (GCF)

Identify and factor out the greatest common factor from the equation. The GCF of the terms in the equation is x.So, we rewrite the equation as:\[x(x^2 + 6x - 7) = 0\]

Factor the Quadratic Expression

We need to factor the quadratic expression \(x^2 + 6x - 7\). Consider the two numbers that multiply to -7 and add up to 6. These numbers are 7 and -1.Thus, we can write the equation as:\[x(x + 7)(x - 1) = 0\]

Solve for x

Set each factor equal to zero and solve for x:1. \(x = 0\)2. \(x + 7 = 0\), solving this gives \(x = -7\)3. \(x - 1 = 0\), solving this gives \(x = 1\)Therefore, the solutions are:\[x = 0,\, x = -7,\, x = 1\]

Key concepts

greatest common factor

When working with polynomials, factoring can often simplify the process of finding solutions. One of the first and most important steps in factoring is identifying the greatest common factor (GCF). The GCF is the highest number that divides all the terms in an equation without any remainder.

For the polynomial given in the exercise, which is \(x^3 + 6x^2 - 7x = 0\), we need to look at each term and find what they all have in common. In this case, each term has at least one 'x' in it. So, the GCF here is 'x'.

Once we've found the GCF, we can factor it out from each term. Factoring out 'x' from \(x^3 + 6x^2 - 7x\) gives us \[ x(x^2 + 6x - 7) = 0 \]. This simplified form makes the equation easier to work with and prepares us for the next step of full factorization.

quadratic expression

After factoring out the greatest common factor, the next task is to handle the quadratic expression. In our exercise, the quadratic expression is \(x^2 + 6x - 7\).

To factor this expression, we need to find two numbers that multiply to give the constant term (-7) and add up to the linear coefficient (6). These two numbers in this case are 7 and -1.

So, we can rewrite the quadratic expression as \(x^2 + 6x - 7 = (x + 7)(x - 1)\). This gives us the fully factored form of the polynomial, \[ x(x + 7)(x - 1) = 0 \].

real solutions

The final step in solving the polynomial equation involves finding the real solutions from the factored form. Recall the factored form we obtained: \[ x(x + 7)(x - 1) = 0 \].

For this product to be zero, at least one of the factors must be zero. We set each factor equal to zero and solve for x:

• \ x = 0 \
• \ x + 7 = 0 \, solving this gives \ x = -7 \
• \ x - 1 = 0 \, solving this gives \ x = 1 \

These solutions are the points where the polynomial intersects the x-axis, also known as the roots of the equation. Hence, the real solutions to the given polynomial \(x^3 + 6x^2 - 7x = 0\) are \ x = 0 \, \ x = -7 \, and \ x = 1 \