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Sagot :
To perform the division [tex]\((5x^4 - x^3 + 45x - 11) \div (5x - 1)\)[/tex], we will follow the process of polynomial long division step by step.
1. Setup the division:
[tex]\[ \frac{5x^4 - x^3 + 45x - 11}{5x - 1} \][/tex]
2. Divide the leading term of the numerator by the leading term of the denominator:
[tex]\[ \frac{5x^4}{5x} = x^3 \][/tex]
3. Multiply the entire divisor by the result obtained in the previous step:
[tex]\[ x^3 \cdot (5x - 1) = 5x^4 - x^3 \][/tex]
4. Subtract this result from the original numerator:
[tex]\[ (5x^4 - x^3 + 45x - 11) - (5x^4 - x^3) = (5x^4 - x^3 + 45x - 11) - 5x^4 + x^3 = 0 + 0 + 45x - 11 = 45x - 11 \][/tex]
5. We now repeat the same process with the new polynomial [tex]\(45x - 11\)[/tex]. Divide the leading term [tex]\(45x\)[/tex] by [tex]\(5x\)[/tex]:
[tex]\[ \frac{45x}{5x} = 9 \][/tex]
6. Multiply the entire divisor by 9:
[tex]\[ 9 \cdot (5x - 1) = 45x - 9 \][/tex]
7. Subtract this result from the current polynomial:
[tex]\[ (45x - 11) - (45x - 9) = (45x - 11) - 45x + 9 = 0 - 2 = -2 \][/tex]
8. Finally, because the remainder [tex]\(-2\)[/tex] is of lower degree than the divisor [tex]\(5x - 1\)[/tex], our process is complete. So, we have:
Quotient: [tex]\(x^3 + 9\)[/tex]
Remainder: [tex]\(-2\)[/tex]
Thus, the division of [tex]\(5x^4 - x^3 + 45x - 11\)[/tex] by [tex]\(5x - 1\)[/tex] gives a quotient of [tex]\(x^3 + 9\)[/tex] and a remainder of [tex]\(-2\)[/tex].
1. Setup the division:
[tex]\[ \frac{5x^4 - x^3 + 45x - 11}{5x - 1} \][/tex]
2. Divide the leading term of the numerator by the leading term of the denominator:
[tex]\[ \frac{5x^4}{5x} = x^3 \][/tex]
3. Multiply the entire divisor by the result obtained in the previous step:
[tex]\[ x^3 \cdot (5x - 1) = 5x^4 - x^3 \][/tex]
4. Subtract this result from the original numerator:
[tex]\[ (5x^4 - x^3 + 45x - 11) - (5x^4 - x^3) = (5x^4 - x^3 + 45x - 11) - 5x^4 + x^3 = 0 + 0 + 45x - 11 = 45x - 11 \][/tex]
5. We now repeat the same process with the new polynomial [tex]\(45x - 11\)[/tex]. Divide the leading term [tex]\(45x\)[/tex] by [tex]\(5x\)[/tex]:
[tex]\[ \frac{45x}{5x} = 9 \][/tex]
6. Multiply the entire divisor by 9:
[tex]\[ 9 \cdot (5x - 1) = 45x - 9 \][/tex]
7. Subtract this result from the current polynomial:
[tex]\[ (45x - 11) - (45x - 9) = (45x - 11) - 45x + 9 = 0 - 2 = -2 \][/tex]
8. Finally, because the remainder [tex]\(-2\)[/tex] is of lower degree than the divisor [tex]\(5x - 1\)[/tex], our process is complete. So, we have:
Quotient: [tex]\(x^3 + 9\)[/tex]
Remainder: [tex]\(-2\)[/tex]
Thus, the division of [tex]\(5x^4 - x^3 + 45x - 11\)[/tex] by [tex]\(5x - 1\)[/tex] gives a quotient of [tex]\(x^3 + 9\)[/tex] and a remainder of [tex]\(-2\)[/tex].
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