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Sagot :
Sure, let's tackle each of these polynomial divisions step by step.
### (a) Divide [tex]\(\left(15 x^4 + 12 x^3 + 9 x^2 + 6 x\right) \)[/tex] by [tex]\( 3 x \)[/tex]:
1. Divide each term of the polynomial by [tex]\(3x\)[/tex].
[tex]\[ \begin{align*} \frac{15 x^4}{3 x} &= 5 x^3, \\ \frac{12 x^3}{3 x} &= 4 x^2, \\ \frac{9 x^2}{3 x} &= 3 x, \\ \frac{6 x}{3 x} &= 2. \end{align*} \][/tex]
2. Combine the results:
[tex]\[ \frac{15 x^4 + 12 x^3 + 9 x^2 + 6 x}{3 x} = 5 x^3 + 4 x^2 + 3 x + 2. \][/tex]
### (c) Divide [tex]\(\left(25 x^5 + 20 x^4 + 15 x^3 + 10 x^2\right) \)[/tex] by [tex]\( 5 x^2 \)[/tex]:
1. Divide each term of the polynomial by [tex]\(5x^2\)[/tex].
[tex]\[ \begin{align*} \frac{25 x^5}{5 x^2} &= 5 x^3, \\ \frac{20 x^4}{5 x^2} &= 4 x^2, \\ \frac{15 x^3}{5 x^2} &= 3 x, \\ \frac{10 x^2}{5 x^2} &= 2. \end{align*} \][/tex]
2. Combine the results:
[tex]\[ \frac{25 x^5 + 20 x^4 + 15 x^3 + 10 x^2}{5 x^2} = 5 x^3 + 4 x^2 + 3 x + 2. \][/tex]
### (b) Divide [tex]\(\left(16 x^4 + 12 x^3 + 8 x^2 + 4 x\right) \)[/tex] by [tex]\( 2 x \)[/tex]:
1. Divide each term of the polynomial by [tex]\(2x\)[/tex].
[tex]\[ \begin{align*} \frac{16 x^4}{2 x} &= 8 x^3, \\ \frac{12 x^3}{2 x} &= 6 x^2, \\ \frac{8 x^2}{2 x} &= 4 x, \\ \frac{4 x}{2 x} &= 2. \end{align*} \][/tex]
2. Combine the results:
[tex]\[ \frac{16 x^4 + 12 x^3 + 8 x^2 + 4 x}{2 x} = 8 x^3 + 6 x^2 + 4 x + 2. \][/tex]
### (d) Divide [tex]\(\left(48 x^5 + 32 x^4 + 16 x^3 + 8 x^2\right) \)[/tex] by [tex]\( 8 x^2 \)[/tex]:
1. Divide each term of the polynomial by [tex]\(8x^2\)[/tex].
[tex]\[ \begin{align*} \frac{48 x^5}{8 x^2} &= 6 x^3, \\ \frac{32 x^4}{8 x^2} &= 4 x^2, \\ \frac{16 x^3}{8 x^2} &= 2 x, \\ \frac{8 x^2}{8 x^2} &= 1. \end{align*} \][/tex]
2. Combine the results:
[tex]\[ \frac{48 x^5 + 32 x^4 + 16 x^3 + 8 x^2}{8 x^2} = 6 x^3 + 4 x^2 + 2 x + 1. \][/tex]
So the detailed solutions are:
1. [tex]\(\frac{15 x^4 + 12 x^3 + 9 x^2 + 6 x}{3 x} = 5 x^3 + 4 x^2 + 3 x + 2\)[/tex]
2. [tex]\(\frac{25 x^5 + 20 x^4 + 15 x^3 + 10 x^2}{5 x^2} = 5 x^3 + 4 x^2 + 3 x + 2\)[/tex]
3. [tex]\(\frac{16 x^4 + 12 x^3 + 8 x^2 + 4 x}{2 x} = 8 x^3 + 6 x^2 + 4 x + 2\)[/tex]
4. [tex]\(\frac{48 x^5 + 32 x^4 + 16 x^3 + 8 x^2}{8 x^2} = 6 x^3 + 4 x^2 + 2 x + 1\)[/tex]
### (a) Divide [tex]\(\left(15 x^4 + 12 x^3 + 9 x^2 + 6 x\right) \)[/tex] by [tex]\( 3 x \)[/tex]:
1. Divide each term of the polynomial by [tex]\(3x\)[/tex].
[tex]\[ \begin{align*} \frac{15 x^4}{3 x} &= 5 x^3, \\ \frac{12 x^3}{3 x} &= 4 x^2, \\ \frac{9 x^2}{3 x} &= 3 x, \\ \frac{6 x}{3 x} &= 2. \end{align*} \][/tex]
2. Combine the results:
[tex]\[ \frac{15 x^4 + 12 x^3 + 9 x^2 + 6 x}{3 x} = 5 x^3 + 4 x^2 + 3 x + 2. \][/tex]
### (c) Divide [tex]\(\left(25 x^5 + 20 x^4 + 15 x^3 + 10 x^2\right) \)[/tex] by [tex]\( 5 x^2 \)[/tex]:
1. Divide each term of the polynomial by [tex]\(5x^2\)[/tex].
[tex]\[ \begin{align*} \frac{25 x^5}{5 x^2} &= 5 x^3, \\ \frac{20 x^4}{5 x^2} &= 4 x^2, \\ \frac{15 x^3}{5 x^2} &= 3 x, \\ \frac{10 x^2}{5 x^2} &= 2. \end{align*} \][/tex]
2. Combine the results:
[tex]\[ \frac{25 x^5 + 20 x^4 + 15 x^3 + 10 x^2}{5 x^2} = 5 x^3 + 4 x^2 + 3 x + 2. \][/tex]
### (b) Divide [tex]\(\left(16 x^4 + 12 x^3 + 8 x^2 + 4 x\right) \)[/tex] by [tex]\( 2 x \)[/tex]:
1. Divide each term of the polynomial by [tex]\(2x\)[/tex].
[tex]\[ \begin{align*} \frac{16 x^4}{2 x} &= 8 x^3, \\ \frac{12 x^3}{2 x} &= 6 x^2, \\ \frac{8 x^2}{2 x} &= 4 x, \\ \frac{4 x}{2 x} &= 2. \end{align*} \][/tex]
2. Combine the results:
[tex]\[ \frac{16 x^4 + 12 x^3 + 8 x^2 + 4 x}{2 x} = 8 x^3 + 6 x^2 + 4 x + 2. \][/tex]
### (d) Divide [tex]\(\left(48 x^5 + 32 x^4 + 16 x^3 + 8 x^2\right) \)[/tex] by [tex]\( 8 x^2 \)[/tex]:
1. Divide each term of the polynomial by [tex]\(8x^2\)[/tex].
[tex]\[ \begin{align*} \frac{48 x^5}{8 x^2} &= 6 x^3, \\ \frac{32 x^4}{8 x^2} &= 4 x^2, \\ \frac{16 x^3}{8 x^2} &= 2 x, \\ \frac{8 x^2}{8 x^2} &= 1. \end{align*} \][/tex]
2. Combine the results:
[tex]\[ \frac{48 x^5 + 32 x^4 + 16 x^3 + 8 x^2}{8 x^2} = 6 x^3 + 4 x^2 + 2 x + 1. \][/tex]
So the detailed solutions are:
1. [tex]\(\frac{15 x^4 + 12 x^3 + 9 x^2 + 6 x}{3 x} = 5 x^3 + 4 x^2 + 3 x + 2\)[/tex]
2. [tex]\(\frac{25 x^5 + 20 x^4 + 15 x^3 + 10 x^2}{5 x^2} = 5 x^3 + 4 x^2 + 3 x + 2\)[/tex]
3. [tex]\(\frac{16 x^4 + 12 x^3 + 8 x^2 + 4 x}{2 x} = 8 x^3 + 6 x^2 + 4 x + 2\)[/tex]
4. [tex]\(\frac{48 x^5 + 32 x^4 + 16 x^3 + 8 x^2}{8 x^2} = 6 x^3 + 4 x^2 + 2 x + 1\)[/tex]
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