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Chapter 8: Problem 66

Solve. $$ x^{4}-x^{3}-6 x^{2}=0[6.2] $$

Short Answer

Expert verified
The solutions are \(x = 0, 3, -2\).

Step by step solution

01

Factor out the greatest common factor

Factor out the greatest common factor (GCF) from the equation. The GCF in this case is \(x^2\). So, we can write:\[ x^2(x^2 - x - 6) = 0 \]
02

Solve for the first factor

Since \(x^2 = 0\) is one of the factors, solve for \(x\): \[ x^2 = 0 \]Taking the square root of both sides, we get:\[ x = 0 \]
03

Factor the quadratic expression

Now, solve for the second part of the equation \(x^2 - x - 6 = 0\). Factor the quadratic expression by finding two numbers that multiply to \(-6\) and add to \(-1\). These numbers are \(-3\) and \(2\). So, we write:\[ x^2 - x - 6 = (x - 3)(x + 2) \]
04

Solve for the second and third factors

Set each factor equal to zero and solve for \(x\): \[ x - 3 = 0 \Rightarrow x = 3 \]\[ x + 2 = 0 \Rightarrow x = -2 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

greatest common factor
To start solving any polynomial equation, the first step is to identify if there's a greatest common factor (GCF) that can be factored out. The GCF is the highest degree of the variable that appears in every term. In the given exercise, each term contains at least an \(x^2\). Taking \(x^2\) out simplifies the equation, making it more manageable to solve.
solving quadratic equations
Once the GCF is factored out, the next step is solving the remaining quadratic equation. A quadratic equation is an equation of the form \(ax^2 + bx + c = 0\). There are multiple methods to solve such equations - factoring is a common one.
In our exercise, after factoring out \(x^2\), we are left with \(x^2 - x - 6 = 0\). This equation can be solved by finding the values of \(x\) that satisfy it.
Sometimes, quadratic equations also require the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For more complex quadratic expressions, these alternative methods can be highly useful.
factoring trinomials
A key aspect of solving our quadratic equation involves factoring trinomials. Trinomials are expressions with three terms. Here we have \(x^2 - x - 6\). To factor this, we need to find two numbers that multiply to the constant term (-6) and add to the linear coefficient (-1). These numbers turn out to be -3 and 2, giving us:
\(x^2 - x - 6 = (x - 3)(x + 2)\)
This factorization transforms the quadratic equation into a simpler form that can be easier to solve. Factoring trinomials is a powerful technique and serves as a fundamental skill in algebra.
roots of polynomials
Finally, let's find the roots of the polynomial. Roots are the values of \(x\) that satisfy the equation (i.e., make the polynomial equal to zero). In our scenario, we solve the factored equation \((x-3)(x+2) = 0\)
Setting each factor equal to zero gives us:
\

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