多項式，已知餘數，求商

 

We are given the function:

$$f(x) = (4x^2 - 2x + 1)(mx + n) + 4x - 9$$

with the conditions:

$$f\left(-\frac{1}{2}\right) = -5, \quad f(1) = -8.$$

We need to determine $m$ and $n$, and then compute $f(2)$.

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Step 1: Compute $f(-\frac{1}{2})$

Substituting $x = -\frac{1}{2}$:

$$4\left(-\frac{1}{2}\right)^2 - 2\left(-\frac{1}{2}\right) + 1 = 4\left(\frac{1}{4}\right) + 1 + 1 = 1 + 1 + 1 = 3.$$
$$f\left(-\frac{1}{2}\right) = (3)(m(-\frac{1}{2}) + n) + 4(-\frac{1}{2}) - 9$$
$$= 3\left(-\frac{m}{2} + n\right) - 2 - 9$$
$$= 3n - \frac{3m}{2} - 11$$

Setting this equal to $-5$:

$$3n - \frac{3m}{2} - 11 = -5$$
$$3n - \frac{3m}{2} = 6$$
$$3n = \frac{3m}{2} + 6$$
$$n = \frac{m}{2} + 2$$

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Step 2: Compute $f(1)$

Substituting $x = 1$:

$$4(1)^2 - 2(1) + 1 = 4 - 2 + 1 = 3.$$
$$f(1) = 3(m(1) + n) + 4(1) - 9$$
$$= 3(m + n) + 4 - 9$$
$$= 3m + 3n - 5$$

Setting this equal to $-8$:

$$3m + 3n - 5 = -8$$
$$3m + 3n = -3$$
$$m + n = -1$$

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Step 3: Solve for $m$ and $n$

We have the system:

1. $n = \frac{m}{2} + 2$
   
2. $m + n = -1$
   

Substituting equation (1) into equation (2):

$$m + \left(\frac{m}{2} + 2\right) = -1$$
$$m + \frac{m}{2} + 2 = -1$$

Multiply everything by 2 to clear the fraction:

$$2m + m + 4 = -2$$
$$3m = -6$$
$$m = -2$$

Substituting $m = -2$ into $n=\frac{m}{2}+\mathrm{2:}$

$$n = \frac{-2}{2} + 2 = -1 + 2 = 1$$

Thus, we have:

$$m = -2, \quad n = 1.$$

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Step 4: Compute $f(2)$

Substituting $x = 2$:

$$4(2)^2 - 2(2) + 1 = 16 - 4 + 1 = 13.$$
$$f(2) = (13)(m(2) + n) + 4(2) - 9$$
$$= 13(-2(2) + 1) + 8 - 9$$
$$= 13(-4 + 1) + 8 - 9$$
$$= 13(-3) + 8 - 9$$
$$= -39 + 8 - 9$$
$$= -40$$

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Final Answer

$$\boxed{{\displaystyle -40}}$$

多項式，x 位移

 

Yes! Let's perform continuous synthetic division step by step until we get the Nested Division Form of

$$3x^3 - 5x^2 - 7x - 4$$

divided by $x-3.$

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Step 1: First Synthetic Division

We divide $3x^3 - 5x^2 - 7x - 4$ using synthetic division.

Set Up the Synthetic Division Table

1. Write the coefficients of the polynomial: $$3, -5, -7, -4$$
   
2. The divisor is $x - 3$, so we use $\mathrm{3\; as\; the\; synthetic\; division\; factor.}$

$$\begin{array}{r|rrrr} 3 & 3 & -5 & -7 & -4 \\ & & 9 & 12 & 15 \\ \hline & 3 & 4 & 5 & 11 \\ \end{array}$$

• Quotient: $3x^2+4x+5$
• Remainder: $11$
  

Thus, we now divide $3x^2+4x+5.$

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Step 2: Second Synthetic Division

Now, divide $3x^2+4x+5.$

$$\begin{array}{r|rrr} 3 & 3 & 4 & 5 \\ & & 9 & 39 \\ \hline & 3 & 13 & 44 \\ \end{array}$$

• Quotient: $3x+13$
• Remainder: $44$

Now, divide $3x + 13$ by $x - 3$.

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Step 3: Third Synthetic Division

Now, divide $3x + 13$ by $x-3.$

$$\begin{array}{r|rr} 3 & 3 & 13 \\ & & 9 \\ \hline & 3 & 22 \\ \end{array}$$

• Quotient: $3$
  
• Remainder: $22$
  

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Step 4: Express in Nested Division Form

Now, we reconstruct the polynomial using the results of the synthetic divisions.

1. The final quotient is 3.

2. The previous quotient was $3x+\mathrm{13,\; which\; we\; write\; as:}$

   $$3(x - 3) + 22$$

3. The next quotient was $3x^2 + 4x + 5$, which we write using the previous step:

   $$3(x - 3)^2 + 22(x - 3) + 44$$

4. Finally, including the remainder:

   $$3(x - 3)^3 + 22(x - 3)^2 + 44(x - 3) + 11$$
   

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Final Answer:

$$\boxed{3(x - 3)^3 + 22(x - 3)^2 + 44(x - 3) + 11}$$

This is the Nested Division Form of $3x^3 - 5x^2 - 7x - 4$ when divided by $x - 3$.

多項式，替換新變數

 

We are given the equation:

$$(x^2 - 3x)^2 - 14(x^2 - 3x) + 40 = 0$$

Step 1: Substituting a New Variable

Let:

$$y = x^2 - 3x$$

Then, our equation transforms into:

$$y^2 - 14y + 40 = 0$$

Step 2: Solve the Quadratic Equation

The quadratic equation:

$$y^2 - 14y + 40 = 0$$

can be solved using the quadratic formula:

$$y = \frac{-(-14) \pm \sqrt{(-14)^2 - 4(1)(40)}}{2(1)}$$ $$y = \frac{14 \pm \sqrt{196 - 160}}{2}$$ $$y = \frac{14 \pm \sqrt{36}}{2}$$ $$y = \frac{14 \pm 6}{2}$$

Step 3: Find Values of $y$

$$y = \frac{14 + 6}{2} = \frac{20}{2} = 10$$
$$y = \frac{14 - 6}{2} = \frac{8}{2} = 4$$

Thus, we have two equations to solve:

$$x^2 - 3x = 10$$
$$x^2 - 3x = 4$$

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Step 4: Solve for $x$

Equation 1: $x^2-3x-10=0$

Factorizing:

$$(x - 5)(x + 2) = 0$$
$$x - 5 = 0 \quad \text{or} \quad x + 2 = 0$$
$$x = 5 \quad \text{or} \quad x = -2$$

Equation 2: $x^2-3x-4=0$

Factorizing:

$$(x - 4)(x + 1) = 0$$
$$x - 4 = 0 \quad \text{or} \quad x + 1 = 0$$
$$x = 4 \quad \text{or} \quad x = -1$$

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Final Answer

$$\boxed{{\displaystyle x=5,-2,4,-1}}$$

多項式，求未定係數

 

Given Equation

$$2x^3 - 4x^2 - 5x + 1 = a(x - 1)(x + 2)(x + 3) + b(x - 1)(x - 2)(x - 3) + c(x + 2)(x - 3)(x + 1)$$

We will substitute three specific values of $x$ to solve for $a, b, c$.

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Step 1: Substituting $x=1$

Since $(x-1)=\mathrm{0,\; the\; terms\; involving }$$a$ and $\mathrm{b vanish,\; leaving:}$

$$f(1) = c(1 + 2)(1 - 3)(1 + 1)$$

Compute $f(1):$

$$2(1)^3 - 4(1)^2 - 5(1) + 1 = 2 - 4 - 5 + 1 = -6$$

Compute the right-hand side:

$$c(3)(-2)(2) = -12c$$

Thus, we solve:

$$-6 = -12c$$
$$c = \frac{1}{2}$$

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Step 2: Substituting $x=-2$

Since $(x+2)=\mathrm{0,\; the\; terms\; involving }$$a$ and $c$ vanish, leaving:

$$f(-2) = b(-2 - 1)(-2 - 2)(-2 - 3)$$

Compute $f(-2)$:

$$2(-2)^3 - 4(-2)^2 - 5(-2) + 1$$
$$= 2(-8) - 4(4) - 5(-2) + 1$$
$$= -16 - 16 + 10 + 1 = -21$$

Compute the right-hand side:

$$b(-3)(-4)(-5) = -60b$$

Thus, we solve:

$$-21 = -60b$$
$$b = \frac{7}{20}$$

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Step 3: Substituting $x=3$

Since $(x-3)=\mathrm{0,\; the\; terms\; involving }$$b$ and $c$ vanish, leaving:

$$f(3) = a(3 - 1)(3 + 2)(3 + 3)$$

Compute $f(3)$:

$$2(3)^3 - 4(3)^2 - 5(3) + 1$$
$$= 2(27) - 4(9) - 15 + 1$$
$$= 54 - 36 - 15 + 1 = 4$$

Compute the right-hand side:

$$a(2)(5)(6) = 60a$$

Thus, we solve:

$$4 = 60a$$
$$a = \frac{1}{15}$$

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Final Answer

$$\boxed{a = \frac{1}{15}, \quad b = \frac{7}{20}, \quad c = \frac{1}{2}}$$

This is the correct set of values for $a, b, c$. 😊

表面張力是什麼？它是怎麼形成的？

 請用台灣國小六年級能理解的語言，說明什麼是表面張力?它是怎麼形成的?

ChatGPT 說：

什麼是表面張力？

表面張力就像是一層看不見的「皮膜」，把水的表面拉緊。想像一下，當你在水面上輕輕放一根針，針雖然比水重，但如果放得很小心，它還是可以浮在水面上，這就是表面張力的效果。

表面張力是怎麼形成的？

水分子之間就像好朋友一樣，會互相拉住對方。在水的內部，每個水分子四周都有其他水分子拉著它，所以它們的力量是平衡的，不會往某個方向移動。

但在水的表面，水分子的上方沒有其他水分子拉住它，只有下面和旁邊的水分子在拉它，這讓水的表面像被拉緊了一樣，就像一張被繃緊的彈簧床一樣，形成一種「皮膜」的感覺。這就是表面張力！

舉個例子：

• 小昆蟲可以在水面上走，像水黽（ㄇㄧㄣˇ）這種小昆蟲，它們的腳很輕，不會弄破水的表面，所以可以站在水上。
• 水滴變成圓形，當你看到雨滴時，會發現水滴大多是圓形的，這是因為表面張力讓水收縮成最小的形狀，而球形剛好是最小的表面積的形狀。

這就是表面張力的概念，是水的一種特別的力量！😊