Question

factorise x^3+x+3
please give right ans ​

Answer 1

Step-by-step explanation:

One of the methods of factorization is using FACTOR & REMAINDER THEOREM:

Factor theorem states that (x-a) is a factor of p(x) if p(a) =0

& remainder theorem states that p(a) = 0 if (x-a) is a factor of p(x)

Given P(x) = x^3 - x² - x + 1

This is a cubic polynomial.. ie, its degree is 3.

So it contains 3 linear factors, the constant term of which will be the factors of the constant term of the polynomial. As while multiplying the constant terms of all 3 linear factors, the product should be +1

So we have +1 or -1 as the constant terms of 3 linear factors.

And as per factor & remainder theorem, (x-a) will be a factor of p(x), if p(a) =0

Here, p(x)= x^3-x²-x+1

& p(1) = 1^3–1²-1+1 = 0

So, ( x-1) is a factor of p(x)

Now if we divide p(x) by (x-1) we get a quadratic polynomial as its quotient. & remainder has to be 0. As (x-1) is a factor of it.

P(x) ÷(x-1) then quotient = (x²-1) which is further factorized into (x+1)& (x-1)

So all 3 factors of p(x) = (x-1)(x+1)(x-1)

Answer 2

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Factorise x³ + 3x - x - 3

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Let p(x)=x³ + 3x² − 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)

∴x³ +3x²

−x−3=(x+1)(x−1)(x+3)

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