Question

What methods can be used to determine the factored form of a polynomial equation, and how can one find the zeros of this equation based on its factored form?

Answer 1

To determine the factored form of a polynomial equation, you can use methods such as factor by grouping, quadratic formula, or dividing by known roots. To find the zeros of the equation based on the factored form, set each factor equal to zero and solve for x.

Step-by-step explanation:

Methods to determine the factored form of a polynomial equation:

1. Factor by grouping: Divide the polynomial into 2 groups and factor out common terms from each group.
2. Use the quadratic formula: For quadratic polynomials, use the formula x = (-b ± √(b^2 - 4ac))/(2a) to find the roots, which gives the factored form.
3. Divide by known roots: If you already know some of the roots, divide the polynomial by (x - root) to find the quotient, which is the factored form.

Finding zeros based on the factored form:

1. Set each factor equal to zero: Equate each factor to zero and solve for x. The solutions are the zeros of the equation.
2. Real and complex zeros: Real zeros are where the graph of the equation crosses the x-axis, while complex zeros occur in conjugate pairs and do not intersect the x-axis.