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What should be subtracted from the polynomial 5 $x^{4}$ − 15 $x^{3}$ + 2 $x^{2}$ so that 5 $x^{3}$ − 1 becomes its factor?

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x^{2}

+ x - 3

x^{2}

+ x - 2

x^{2}

+ x - 3

x^{2}

+ x - 2

answer is A.

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Detailed Solution

We know that if x is a factor of a polynomial p(x), then p(x) is exactly divisible by x.
This means that when p(x) is divided by x, the remainder should be equal to 0.
Now, we know that the division algorithm states that if p(x) and g(x) are two polynomials where g(x) ≠ 0, then there are two polynomials q(x) and r(x) such that p(x) = q(x) × g(x) + r(x). In other words, Dividend = Quotient × Divisor + Remainder.
Rewriting the division algorithm, we get
⇒ p(x) − r(x) = q(x) × g(x)
⇒

\frac{p (x) - r (x)}{g (x)}

= q(x)
We need to use a long division method to divide 5

x^{4}

− 15

x^{3}

+ 2

x^{2}

by 5

x^{3}

− 1.
The remainder obtained will be the polynomial/number that should be subtracted from the polynomial 5

x^{4}

−15

x^{3}

x^{2}

so that 5

x^{3}

− 1 becomes its factor.
Now, let us use a long division method to divide 5

x^{4}

− 15

x^{3}

+ 2

x^{2}

by 5

x^{3}

− 1 .
Therefore, we get

https://www.vedantu.com/question-sets/e9f14ffa-745d-4d16-baa3-849ce35b422f7988041280461724207.png

Therefore, 2

x^{2}

+ x −3 should be subtracted from 5

x^{4}

− 15

x^{3}

+ 2

x^{2}

so that 5

x^{3}

− 1 becomes its factor.
We can also verify our answer by subtracting 2

x^{2}

+ x −3 from 5

x^{4}

− 15

x^{3}

+ 2

x^{2}

.
⇒ 5

x^{4}

− 15

x^{3}

+ 2

x^{2}

− (2

x^{2}

+ x −3) = 5

x^{4}

− 15

x^{3}

+ 2

x^{2}

− 2

x^{2}

− x + 3
⇒ 5

x^{4}

− 15

x^{3}

+ 2

x^{2}

− (2

x^{2}

+ x −3) = 5

x^{4}

− 15

x^{3}

− x + 3
⇒ 5

x^{4}

− 15

x^{3}

+ 2

x^{2}

− (2

x^{2}

+ x −3) = 5

x^{3}

(x − 3) − 1(x − 3)
⇒ 5

x^{4}

− 15

x^{3}

+ 2

x^{2}

− (2

x^{2}

+ x −3) = (5

x^{3}

− 1) (x − 3)
Therefore, we can observe that when 2

x^{2}

+ x −3 is subtracted from 5

x^{4}

− 15

x^{3}

+ 2

x^{2}

, 5

x^{3}

− 1 becomes a factor of 5

x^{4}

− 15

x^{3}

+ 2

x^{2}

.
Hence, option (1) is correct.

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What should be subtracted from the polynomial 5x4 − 15x3 + 2x2 so that 5x3 − 1 becomes its factor?

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What should be subtracted from the polynomial 5 $x^{4}$ − 15 $x^{3}$ + 2 $x^{2}$ so that 5 $x^{3}$ − 1 becomes its factor?