Question

Solve. a) \(x(x+3)(x-4)=0\) b) \((x-3)(x-5)(x+1)=0\) c) \((2 x+4)(x-3)=0\)

Short answer

a) \(x=0, -3, 4\) b) \(x=3, 5, -1\) c) \(x=-2, 3\)

Step-by-step solution

Solve for each factor equal to zero for part (a)

Given the equation \(x(x+3)(x-4)=0\), set each factor equal to zero and solve for \(x\): 1. \(x=0\) 2. \(x+3=0\Rightarrow x=-3\) 3. \(x-4=0\Rightarrow x=4\)

Combine solutions for part (a)

The solutions for part (a) are \(x=0\), \(x=-3\), and \(x=4\).

Solve for each factor equal to zero for part (b)

Given the equation \((x-3)(x-5)(x+1)=0\), set each factor equal to zero and solve for \(x\): 1. \(x-3=0\Rightarrow x=3\) 2. \(x-5=0\Rightarrow x=5\) 3. \(x+1=0\Rightarrow x=-1\)

Combine solutions for part (b)

The solutions for part (b) are \(x=3\), \(x=5\), and \(x=-1\).

Solve for each factor equal to zero for part (c)

Given the equation \((2x+4)(x-3)=0\), set each factor equal to zero and solve for \(x\): 1. \(2x+4=0\Rightarrow 2x=-4\Rightarrow x=-2\) 2. \(x-3=0\Rightarrow x=3\)

Combine solutions for part (c)

The solutions for part (c) are \(x=-2\), and \(x=3\).

Key concepts

Factoring

Factoring is a method used to break down complex polynomial equations into simpler parts. This process converts a polynomial into a product of smaller polynomials or factors. These factors are usually easier to work with and solve. For example, in the equation \(x(x+3)(x-4)=0\), the polynomial is already factored into three parts: \(x\), \(x+3\), and \(x-4\).
Factoring is instrumental when solving polynomial equations because it can simplify the equation significantly. By expressing the polynomial as a product of factors, we can individually solve each factor to find the roots of the original equation.

Roots of Polynomial

The roots of a polynomial are the values of \(x\) that make the polynomial equal to zero. For instance, consider the polynomial \(x(x+3)(x-4)=0\). Setting each factor equal to zero helps us find the roots:
1. \( x = 0\)
2. \( x + 3 = 0 \Rightarrow x = -3\)
3. \( x - 4 = 0 \Rightarrow x = 4\)
So, the roots of the polynomial \(x(x+3)(x-4)=0\) are 0, -3, and 4.
Finding the roots is crucial because they provide the solutions to the polynomial equation. When you find all the roots, you have effectively solved the equation.

Zero-Product Property

The Zero-Product Property is a fundamental principle used in solving polynomial equations that have been factored. It states that if the product of several factors is zero, then at least one of the factors must be zero. This property is expressed as follows:
If \(a\cdot b = 0\), then either \(a = 0\) or \(b = 0\) (or both).
For example, to solve \((2x+4)(x-3)=0\):
1. Set each factor equal to zero:
\(2x + 4 = 0\) and \(x - 3 = 0\)
2. Solve each equation:
\(2x = -4 \Rightarrow x = -2\)
\(x = 3\)
Using the Zero-Product Property simplifies the process by allowing us to break down the polynomial equation into more manageable parts and solve for the roots individually.

Quadratic Equations

Quadratic equations are polynomials of degree two, typically expressed in the form \(ax^2 + bx + c = 0\). These equations can be solved using various methods, including factoring, completing the square, and the quadratic formula. For instance, the quadratic equation \((x-3)(x-5)=0\) can be solved by factoring:
1. This equation is already factored: \((x-3)\) and \((x-5)\)
2. Set each factor to zero and solve for \(x\):
\(x-3=0 \Rightarrow x=3\)
\(x-5=0 \Rightarrow x=5\)
Therefore, the roots of the quadratic equation are \(x=3\) and \(x=5\).
Quadratic equations are prevalent in various fields of study and applications, making understanding how to solve them essential.