What should be subtracted from the polynomial $p (x) = 3 x^{3} + 4 x^{2} + 5 x - 13$ , so that the resulting polynomials divisible by $x^{2} - 2 x + 3$ ?

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Solution

We have,

Polynomial $p (x) = 3 x^{3} + 4 x^{2} + 5 x - 13$

Any division statement can be rewriiten in the form of
Dividend $=$ Divisor $\times$ Quotient + Remainder.
When the remainder is zero, the dividend is said to be the multiple of the divisor.

When the remainder is not equal to zero, and if it is subtracted from the dividend, then the dividend would become a multiple of divisor.

Dividend $= 3 x^{3} + 4 x^{2} + 5 x - 13$

Divisor $= x^{2} - 2 x + 3$

Divide the dividend by divisor, to get

Quotient $= 3 x + 10$

Reminder $= 16 x - 43$

Hence, On subtracting $16 x - 43$ from the dividend, it will be exactly divisible by the given divisor.

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Similar questions

What must be subtracted from polynomial $f (x) = x^{4} + 2 x^{3} - 13 x^{2} - 12 x + 21$ so that the resulting polynomial is exactly divisible by $x^{2} - 4 x + 3$ ?

Q. What should be subtracted from

3 x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 3

, so that the resulting polynomial is exactly divisible by

(3 x + 2)

Q. What the number should be subtracted from

{2 x}^{3} - {5 x}^{2} + 5 x

so that the resulting polynomial has

2 x - 3

as a factor?

The expression that should be subtracted from the polynomial f(x) = x⁴ + 2x³ -13x² - 12x + 21 so that the resulting polynomial is exactly divisible by g(x) = x² – 4x + 3 is

Q. What must be subtracted from

8 x^{4} + 14 x^{3} - 2 x^{2} + 7 x - 8

so that the resulting polynomial is exactly divisible by

4 x^{2} + 3 x - 2

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What should be subtracted from the polynomial p(x)=3x3+4x2+5x−13, so that the resulting polynomials divisible by x2−2x+3?

What should be subtracted from the polynomial $p (x) = 3 x^{3} + 4 x^{2} + 5 x - 13$ , so that the resulting polynomials divisible by $x^{2} - 2 x + 3$ ?