Roots of polynomial equations · 多项式方程的根
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| quadratic/kwɒˈdrætɪk/ | 二次 | èr cì |
| root/ruːt/ | 根 | gēn |
| coefficient/ˌkəʊɪˈfɪʃənt/ | 系数 | xì shù |
| cubic/ˈkjuːbɪk/ | 三次 | sān cì |
| substitution/ˌsʌbstɪˈtjuːʃn/ | 代换 | dài huàn |
The secret hidden in every equation
- Every quadratic 二次 equation hides two numbers — its roots 根. You can find their sum and product without solving the equation at all.
- These relationships between roots and coefficients 系数 unlock problems that would otherwise require messy algebra.
每个方程里隐藏的秘密
- 每个二次方程都藏着两个数——它的根。你能在完全不解方程的情况下找到它们的和与积。
- 这些根与系数之间的关系解锁了原本需要凌乱代数的问题。
Roots of a quadratic
- For $ax^2 + bx + c = 0$ with roots $\alpha, \beta$:
Worked example. $2x^2 - 7x + 3 = 0$. Sum of roots $= \dfrac{7}{2}$, product $= \dfrac{3}{2}$. Check: roots are $3$ and $\dfrac{1}{2}$: sum $= 3.5$ ✓, product $= 1.5$ ✓.
The roots of a quadratic and how they relate to its coefficients
二次方程的根
- 对 $ax^2 + bx + c = 0$,根为 $\alpha, \beta$:
算例。 $2x^2 - 7x + 3 = 0$。根的和 $= \dfrac{7}{2}$,积 $= \dfrac{3}{2}$。检查:根是 $3$ 和 $\dfrac{1}{2}$:和 $= 3.5$ ✓,积 $= 1.5$ ✓。
二次方程的根以及它们与系数的关系
Roots of a polynomial · 一个多项式的根
y = ax³ + bx² + cx + d
The roots · 根 are where the curve meets the x-axis — their sum and product link to the coefficients. · 根是曲线与 x 轴相交的地方——它们的和与积联系到系数。
For x² − 5x + 6 = 0, what is the sum of the roots α + β = −b/a? · 对 x² − 5x + 6 = 0,根的和 α + β = −b/a 是多少?
α + β = −b/a = −(−5)/1 = 5.
For x² − 5x + 6 = 0, what is the product αβ = c/a? · 对 x² − 5x + 6 = 0,积 αβ = c/a 是多少?
αβ = c/a = 6/1 = 6.
Roots of a cubic 三次
- For a cubic $ax^3 + bx^2 + cx + d = 0$ with roots $\alpha, \beta, \gamma$:
- $\sum\alpha = -\dfrac{b}{a}$, $\sum\alpha\beta = \dfrac{c}{a}$, $\alpha\beta\gamma = -\dfrac{d}{a}$.
Watch the signs. The sum of roots is $-\dfrac{b}{a}$ (negative), and the product of three roots is $-\dfrac{d}{a}$ (also negative). Getting the signs wrong is the most common error.
三次方程的根
- 对一个三次方程 $ax^3 + bx^2 + cx + d = 0$,根为 $\alpha, \beta, \gamma$:
- $\sum\alpha = -\dfrac{b}{a}$,$\sum\alpha\beta = \dfrac{c}{a}$,$\alpha\beta\gamma = -\dfrac{d}{a}$。
注意符号。 根的和是 $-\dfrac{b}{a}$(负的),三个根的积是 $-\dfrac{d}{a}$(也是负的)。把符号弄错是最常见的错误。
For x³ − 6x² + 11x − 6 = 0, the sum of roots Σα = −b/a. Find it. · 对 x³ − 6x² + 11x − 6 = 0,根的和 Σα = −b/a。求它。
Σα = −(−6)/1 = 6. (Roots are 1, 2, 3: sum = 6.) · Σα = −(−6)/1 = 6。(根是 1、2、3:和 = 6。)
For ax³ + bx² + cx + d = 0, the product of roots αβγ = d/a. · 对 ax³ + bx² + cx + d = 0,根的积 αβγ = d/a。
The product of three roots is αβγ = −d/a (negative), not d/a. · 三个根的积是 αβγ = −d/a(负的),不是 d/a。
Match each root relationship to its formula (leading coefficient a). · 把每个根的关系与它的公式配对(首项系数 a)。
Quadratic: α+β = −b/a, αβ = c/a. Cubic: Σα = −b/a, αβγ = −d/a (sign alternates with degree). · 二次:α+β = −b/a,αβ = c/a。三次:Σα = −b/a,αβγ = −d/a(符号随次数交替)。
Transforming roots
- To find an equation whose roots are changed simply, use a substitution 代换 (e.g. $w = \alpha + 1$).
- Example: $x^2 - 5x + 6 = 0$ ($\alpha+\beta=5$, $\alpha\beta=6$) → roots $\alpha+1,\beta+1$ give sum $7$, product $12$, so $x^2 - 7x + 12 = 0$.
变换根
- 要找一个根被简单改变的方程,用一个代换(substitution,例如 $w = \alpha + 1$)。
- 例子:$x^2 - 5x + 6 = 0$($\alpha+\beta=5$,$\alpha\beta=6$)→ 根 $\alpha+1,\beta+1$ 给出和 $7$、积 $12$,所以 $x^2 - 7x + 12 = 0$。
With α + β = 5 and αβ = 6, what is the product (α+1)(β+1)? · 已知 α + β = 5 而 αβ = 6,积 (α+1)(β+1) 是多少?
(α+1)(β+1) = αβ + α + β + 1 = 6 + 5 + 1 = 12.
Worked example — finding $\alpha^2 + \beta^2$
- Given $x^2 - 5x + 6 = 0$: $\alpha + \beta = 5$, $\alpha\beta = 6$.
- $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = 25 - 12 = 13$.
- Sums like $\sum\alpha$ are symmetric functions of the roots.
算例——求 $\alpha^2 + \beta^2$
- 给定 $x^2 - 5x + 6 = 0$:$\alpha + \beta = 5$,$\alpha\beta = 6$。
- $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = 25 - 12 = 13$。
- 像 $\sum\alpha$ 这样的和是根的对称函数(symmetric functions)。
For x² − 5x + 6 = 0, α² + β² = (α+β)² − 2αβ. Find it. · 对 x² − 5x + 6 = 0,α² + β² = (α+β)² − 2αβ。求它。
(α+β)² − 2αβ = 25 − 12 = 13.
You've got it
- quadratic: $\alpha+\beta = -\dfrac{b}{a}$, $\alpha\beta = \dfrac{c}{a}$
- cubic: $\sum\alpha = -\dfrac{b}{a}$, $\sum\alpha\beta = \dfrac{c}{a}$, $\alpha\beta\gamma = -\dfrac{d}{a}$
- transform roots with a substitution
你掌握了
- 二次:$\alpha+\beta = -\dfrac{b}{a}$,$\alpha\beta = \dfrac{c}{a}$
- 三次:$\sum\alpha = -\dfrac{b}{a}$,$\sum\alpha\beta = \dfrac{c}{a}$,$\alpha\beta\gamma = -\dfrac{d}{a}$
- 用一个代换变换根