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Sagot :
Answer:
[tex](2\, x^{3} - 3\, x^{2} - 18\, x - 8) / (x - 4) = (2\, x^{2} + 5\, x + 2)[/tex].
Step-by-step explanation:
The leading term of a one-variable polynomial refers to the term with the highest power of the variable. For example, in the polynomial [tex](2\, x^{3} - 3\, x^{2} - 18\, x - 8)[/tex], [tex]x[/tex] is the variable, and [tex]2\, x^{3}[/tex] is the leading term. The power of the leading term [tex]2\, x^{3}\![/tex] is [tex]3[/tex].
The numerator is [tex](2\, x^{3} - 3\, x^{2} - 18\, x - 8)[/tex] while the denominator is [tex](x - 4)[/tex].
- Leading term of the numerator: [tex]2\, x^{3}[/tex].
- Leading term of the denominator: [tex]x[/tex].
Divide the leading term of the numerator by the leading term of the denominator to find the next term of the quotient. In this case, [tex](2\, x^{3}) / (x) = 2\, x^{2}[/tex].
Multiply the denominator by the quotient that was just found. In this case, [tex]2\, x^{2}\, (x - 4) = 2\, x^{3} - 8\, x^{2}[/tex]. Subtract this product from the current numerator to find the next numerator:
[tex]\begin{aligned} & (2\, x^{3} - 3\, x^{2} - 18\, x - 8) - (2\, x^{3} - 8\, x^{2}) \\ =\; & 2\, x^{3} - 3\, x^{2} - 18\, x - 8 - 2\, x^{3} + 8\, x^{2}\\=\; & 5\, x^{2} - 18\, x - 8\end{aligned}[/tex].
Thus, the next numerator would be [tex](5\, x^{2} - 18\, x - 8)[/tex]. The denominator stays
the same.
Repeat the steps above until the power of the numerator is less than that of the denominator.
The numerator is [tex](5\, x^{2} - 18\, x - 8)[/tex] while the denominator is still [tex](x - 4)[/tex].
- Leading term of the numerator: [tex]5\, x^{2}[/tex].
- Leading term of the denominator: [tex]x[/tex].
Next term of the quotient: [tex](5\, x^{2}) / (x) = 5\, x[/tex]. Add that term to the quotient. The quotient is now [tex](2\, x^{2} + 5\, x)[/tex].
The next numerator should be:
[tex]\begin{aligned} & (5\, x^{2} - 18\, x - 8) - 5\, x \, (x - 4) \\ =\; & (5\, x^{2} - 18\, x - 8) - (5\, x^{2} - 20) \\ =\; & 2\, x - 8\end{aligned}[/tex].
The numerator is now [tex](2\, x - 8)[/tex] while the denominator continues to be the same.
- Leading term of the numerator: [tex]2\, x[/tex].
- Leading term of the denominator: [tex]x[/tex].
Next term of the quotient: [tex](2\, x) / (x) = 2[/tex]. Add that term to the quotient to get [tex](2\, x^{2} + 5\, x + 2)[/tex].
The next numerator should be:
[tex]\begin{aligned} & (2\, x - 8) - 2\, (x - 4) \\ =\; & (2\, x - 8) - (2\, x - 8) \\ =\; & 0 \end{aligned}[/tex].
The numerator is now a constant. The power of [tex]x[/tex] is the current numerator would be [tex]0[/tex]. The power of [tex]x\![/tex] in the denominator continues to be [tex]1[/tex]. Thus, the power of the numerator is now less than that of the denominator.
The quotient is now the required quotient, with the "numerator" being the remainder of the division. That is:
[tex](2\, x^{3} - 3\, x^{2} - 18\, x - 8) = (2\, x^{2} + 5\, x + 2)\, (x - 4) + 0[/tex].
[tex](2\, x^{3} - 3\, x^{2} - 18\, x - 8) = (2\, x^{2} + 5\, x + 2)\, (x - 4)[/tex].
Equivalently:
[tex](2\, x^{3} - 3\, x^{2} - 18\, x - 8) / (x - 4) = (2\, x^{2} + 5\, x + 2)[/tex].
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