91Ó°ÊÓ

Factoring polynomials is a crucial step in solving polynomial equations. It involves breaking down a complex polynomial into a product of simpler polynomials, which can make equations easier to solve.

To factor a polynomial, look for factors common to all terms in the polynomial. In our exercise, we started with the polynomial \(10x^3 - 30x^2 + 20x\).

By factoring out the greatest common factor (GCF), which is 10x, we simplified the polynomial to \(10x(x^2 - 3x + 2)\).

Once the polynomial is factored, solving the equation becomes straightforward by setting each factor to zero and solving the simpler equations that result.

The greatest common factor (GCF) is the largest number and/or variable that divides all terms in an expression without leaving a remainder. Identifying the GCF is the first step in factoring polynomials.

In our problem, the terms of the polynomial are \(10x^3, -30x^2,\) and \(20x\). To find the GCF:

• First, find the highest number that divides 10, 30, and 20—which is 10.
• Secondly, look for the variable part, which in this case is 'x' common in all terms.

By combining both, the GCF is 10x. Using the GCF, we factored the polynomial into \(10x(x^2 - 3x + 2)\).

Quadratic equations are polynomials of degree 2. They take the form \(ax^2 + bx + c = 0\), where a, b, and c are constants.

In our exercise, the polynomial inside the parentheses, \(x^2 - 3x + 2\), is a quadratic equation.

To factor the quadratic equation, we look for two numbers that multiply to 'c' (the constant term) and add up to 'b' (the coefficient of the linear term). Here, the numbers are -1 and -2 because:

• (-1) * (-2) = 2 (matches the constant term)
• (-1) + (-2) = -3 (matches the middle term)

Thus, \(x^2 - 3x + 2\) factors to \((x - 1)(x - 2)\). Now, we can find solutions by setting each factor to zero and solving: \(x - 1 = 0\) gives \(x = 1\) and \(x - 2 = 0\) gives \(x = 2\). This way, we solve the quadratic equations within the larger polynomial equation.