Question

1. If (x - 1) and (x + 3) are factors of x' + 3x2 - x – 3, find the other factor.

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Answer 1

Answer:

Let p(x)=x

3

+3x

2

−x−3

p(x) is a cubic polynomial, so it may have three linear factors.

The constant term is -3. The factors of -3 are -1, 1, -3 and 3.

p(−1)=(−1)

3

+3(−1)

2

−(−1)−3=−1+3+1−3=0

∴(x+1) is a factor of p(x).

p(1)=(1)

3

+3(1)

2

−1−3=1+3−1−3=0

∴(x−1) is a factor of p(x).

p(−3)=(−3)

3

+3(−3)

2

−(−3)−3=−27+27+3−3=0

∴(x+3) is a factor of p(x).

The three factors of p(x) are (x + 1), (x - 1) and (x - 3)

∴x

3

+3x

2

−x−3=(x+1)(x−1)(x+3).