Zeros of polynomials (multiplicity)  Polynomial graphs  Algebra 2  Khan Academy  Summary and Q&A
TL;DR
Learn how to determine the factors of a polynomial by analyzing the roots and exponents in factored form.
Questions & Answers
Q: Why is factored form useful in analyzing polynomial roots?
Factored form helps identify the roots, as the xvalues that make the polynomial equal to zero can be determined by analyzing the factors.
Q: How does the presence or absence of sign changes affect the exponents in factored form?
Sign changes around a root indicate an odd exponent in the corresponding factor, while the absence of a sign change suggests an even exponent.
Q: How can we determine the correct factors when given multiplechoice options?
By comparing the roots and exponents in the options with the sign changes and expected exponents, the factors consistent with the polynomial's roots can be identified.
Q: How can we determine the factors if the graph touches the xaxis but does not have a sign change?
If there is no sign change, the exponent in the corresponding factor should be even, as the graph only touches the xaxis briefly before returning to its initial direction.
Summary & Key Takeaways

Factored form of a polynomial is useful for determining its roots, which are the xvalues that make the polynomial equal to zero.

The odd or even exponents in the factors of a polynomial depend on the presence or absence of sign changes around the corresponding roots.

By analyzing the roots and exponents in multiplechoice options, the correct factors of a polynomial can be identified.