Equivalent Representations of Expressions · 表达式的等价形式
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| equivalent form/ɪˈkwɪvələnt fɔːm/ | 等价形式 | děng jià xíng shì |
| factoring/ˈfæktərɪŋ/ | 因式分解 | yīn shì fēn jiě |
| zeros/ˈzɪərəʊz/ | 零点 | líng diǎn |
| polynomial long division/ˌpɒlɪˈnəʊmɪəl lɒŋ dɪˈvɪʒn/ | 多项式长除法 | duō xiàng shì zhǎng chú fǎ |
| binomial theorem/baɪˈnəʊmɪəl ˈθɪərəm/ | 二项式定理 | èr xiàng shì dìng lǐ |
Same function, many disguises
- The rule $y = (x+1)(x-3)$ and the rule $y = x^2 - 2x - 3$ look different.
- Yet they draw the exact same parabola — they are the same function.
- One expression can be written in several equal-looking-but-equivalent ways.
- The skill is choosing the disguise that shows the feature you want.
同一个函数,多种伪装
- 式子 $y = (x+1)(x-3)$ 和式子 $y = x^2 - 2x - 3$ 看起来不一样。
- 可它们画出完全相同的抛物线——它们是同一个函数。
- 一个表达式可以写成好几种看起来不同、但等价的形式。
- 关键技能是选出能显示你想要特征的那种伪装。
Factored and expanded forms
- An equivalent form 等价形式 is a rewrite that gives the same output for every input.
- Factoring 因式分解 turns $x^2 - 2x - 3$ into $(x+1)(x-3)$; expanding does the reverse.
- The factored form shows the zeros 零点 at once: set each factor to zero → $x = -1, 3$.
- The expanded form shows the leading term, and so the end behavior.
因式形式与展开形式
- 等价形式(equivalent form)是一种改写,对每个输入都给出相同的输出。
- 因式分解(factoring)把 $x^2 - 2x - 3$ 变成 $(x+1)(x-3)$;展开则相反。
- 因式形式立刻显示零点(zeros):令每个因式为零 → $x = -1, 3$。
- 展开形式显示首项,从而显示末端行为。
One parabola, written two ways · 一条抛物线,两种写法
y = (x+1)(x−3) = x² − 2x − 3
This curve is y = (x+1)(x−3) and also y = x² − 2x − 3. The factored form shows the zeros −1 and 3; the expanded form shows the leading term. · 这条曲线既是 y = (x+1)(x−3),也是 y = x² − 2x − 3。因式形式显示零点 −1 和 3;展开形式显示首项。
Pick the form that reveals the feature
- Want the zeros or x-intercepts? Use the factored form.
- Want the y-intercept? Any form works — just evaluate at $x = 0$.
- Want the end behavior? Expand and read the leading term.
- Choosing wisely saves work; the answer is the same either way.
选出能揭示特征的形式
- 想要零点或 x 轴交点?用因式形式。
- 想要 y 轴交点?任何形式都行——在 $x = 0$ 处求值即可。
- 想要末端行为?展开并读出首项。
- 明智地选择能省力;无论用哪种,答案都相同。
Which form of a polynomial most quickly reveals its zeros? · 多项式的哪种形式最快显示出它的零点?
In factored form $(x-a)(x-b)$, the zeros $a$ and $b$ are read off instantly by setting each factor to zero. · 在因式形式 $(x-a)(x-b)$ 中,令每个因式为零就能立刻读出零点 $a$ 和 $b$。
Which form most quickly reveals the end behavior? · 哪种形式最快显示末端行为?
The expanded form shows the leading term directly, and the leading term controls end behavior. · 展开形式直接显示首项,而首项决定末端行为。
Long division for rational expressions
- Polynomial long division 多项式长除法 rewrites $\frac{\text{top}}{\text{bottom}}$ as a quotient plus a remainder over the bottom.
- That form exposes the horizontal or slant asymptote directly — it is the quotient.
- It is the rational-function version of turning $\frac{7}{2}$ into $3 + \frac{1}{2}$.
- Divide when you need the asymptotic behavior of a fraction.
有理表达式的长除法
- 多项式长除法(polynomial long division)把 上/下 改写成一个商加上余数除以下面。
- 那种形式直接暴露水平或斜渐近线——它就是那个商。
- 这是把 $\frac{7}{2}$ 变成 $3 + \frac{1}{2}$ 的有理函数版本。
- 当你需要一个分数的渐近行为时,就做除法。
Polynomial long division rewrites a rational expression as a quotient plus a ____ over the denominator. · 多项式长除法把有理表达式改写成一个商加上一个____除以分母。
$\frac{\text{top}}{\text{bottom}} = \text{quotient} + \frac{\text{remainder}}{\text{bottom}}$ — the form that reveals asymptotic behavior. · 写成"商 + 余数/分母"——这种形式能揭示渐近行为。
Binomial theorem for powers
- Expanding $(a+b)^n$ by hand gets slow as $n$ grows.
- The binomial theorem 二项式定理 gives every term of the expansion using a fixed pattern of coefficients.
- For example $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$.
- It is the quick route from a compact power to its expanded form.
用二项式定理展开幂
- 手动展开 $(a+b)^n$ 会随着 $n$ 变大而变慢。
- 二项式定理(binomial theorem)用一套固定的系数模式,给出展开式的每一项。
- 例如 $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$。
- 它是从一个紧凑的幂到它展开形式的捷径。
The binomial theorem helps you expand expressions of the form… · 二项式定理帮你展开形如……的表达式。
The binomial theorem gives the expanded terms of $(a+b)^n$ without multiplying it out by hand. · 二项式定理给出 $(a+b)^n$ 展开后的各项,不必手动一项项相乘。
Select all · 所有 true statements about choosing a form. · 选出关于选择形式的所有正确说法。
Equivalent forms never change the output — they only make different features easy to see. · 等价形式从不改变输出——它们只是让不同的特征更容易看见。
Two forms are equivalent only on their common domain. Cancelling $\frac{(x-1)(x+2)}{x-1}$ to $x+2$ changes the domain — the forms match everywhere except the hole. Equivalent means "same output where both are defined", not "identical everywhere".
两种形式只有在它们的共同定义域上才等价。把 $\frac{(x-1)(x+2)}{x-1}$ 化简为 $x+2$ 改变了定义域——两种形式在除空心外处处相同。等价的意思是"在两者都有定义之处输出相同",而不是"处处完全一样"。
Two equivalent forms give the same output for every input in their common domain. · 两种等价形式在它们共同的定义域内,对每个输入都给出相同的输出。
That is what "equivalent" means — identical values everywhere both are defined. Watch for domain gaps a cancellation may hide. · 这正是"等价"的含义——在两者都有定义之处值完全相同。要留意约分可能隐藏的定义域缺口。
Rewrite $x^2 - 5x + 6$ to find its zeros.
- Factor: $x^2 - 5x + 6 = (x-2)(x-3)$.
- Set each factor to zero: $x = 2$ and $x = 3$.
- The expanded form hid these; the factored form hands them over instantly.
改写 $x^2 - 5x + 6$ 来求它的零点。
- 因式分解:$x^2 - 5x + 6 = (x-2)(x-3)$。
- 令每个因式为零:$x = 2$ 和 $x = 3$。
- 展开形式把它们藏了起来;因式形式则立刻交出答案。
An equivalent form rewrites an expression without changing its output. Factoring exposes the zeros; expanding exposes the leading term; polynomial long division exposes an asymptote; the binomial theorem expands $(a+b)^n$. Pick the form that reveals what you need.
等价形式在不改变输出的前提下改写表达式。因式分解暴露零点;展开暴露首项;多项式长除法暴露渐近线;二项式定理展开 $(a+b)^n$。选出能揭示你所需内容的那种形式。