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Using long division, determine if the following polynomial divides evenly for a divisor of x+3x3−5x2+10x−15.
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You want to divide x^3 − 5x^2 + 10x − 15 by x + 3.
Long division:
- x^3 ÷ x = x^2 → multiply: x^2(x+3)=x^3+3x^2. Subtract → −8x^2 + 10x.
- (−8x^2) ÷ x = −8x → multiply: −8x(x+3)=−8x^2−24x. Subtract → 34x − 15.
- 34x ÷ x = 34 → multiply: 34(x+3)=34x+102. Subtract → remainder −117.
Quotient = x^2 − 8x + 34, remainder = −117.
Since the remainder ≠ 0, x + 3 does not divide x^3 − 5x^2 + 10x − 15 evenly. (By the Remainder Theorem, f(−3)=−117.)
Long division:
- x^3 ÷ x = x^2 → multiply: x^2(x+3)=x^3+3x^2. Subtract → −8x^2 + 10x.
- (−8x^2) ÷ x = −8x → multiply: −8x(x+3)=−8x^2−24x. Subtract → 34x − 15.
- 34x ÷ x = 34 → multiply: 34(x+3)=34x+102. Subtract → remainder −117.
Quotient = x^2 − 8x + 34, remainder = −117.
Since the remainder ≠ 0, x + 3 does not divide x^3 − 5x^2 + 10x − 15 evenly. (By the Remainder Theorem, f(−3)=−117.)
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