韦达公式与因式分解

Introduction: Combining Identities

Vieta's Formulas

In $x^2-Sx+P=(x-y)(x-z)$, what do $S$ and $P$ represent?

Derivation of Vieta's Formulas

When factoring the quadratic trinomial $x^2+7x+12$ with the variant of Vieta's formulas, what signs should the factors take?

Three-Variable Square and Cube Identities

When $x+y+z=0$, what does $x^3+y^3+z^3$ equal?

Handling Symmetric Expressions

When $x+y+z=0$ (not all zero), what is the value of $\dfrac{x^2+y^2+z^2}{xy+yz+zx}$?

Worked Example: Factoring a Quadratic Trinomial

Use the variant of Vieta's formulas to factor $x^2+7x+12$ as the product of two linear factors (no spaces).

Factor $x^2-x-6$ as the product of two linear factors (no spaces).

Given $a+b+c=0$ and $abc=5$, find the value of $a^3+b^3+c^3$. Enter an integer.

What are Vieta's formulas and their variant? How are $S$ and $P$ determined?

Explain in your own words: why does $x^3+y^3+z^3=3xyz$ hold when $x+y+z=0$? What is the key step of the derivation?

Comprehensive Check

Correcting Common Errors and Lesson Summary