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Q.

If (y−3) is a factor of $y^{3} + {2 y}^{2} - 9 y - 18$ . Then the other two factors are and .

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Detailed Solution

Concept - If (y−3) is a factor of

y^{3} + {2 y}^{2} - 9 y - 18

Then the other two factors are

(y + 2)

and

(y + 3)

.
When only one factor is given, polynomial division can be used to identify the remaining factors. The additional two elements can be discovered by first converting the third degree polynomial into a second degree polynomial.
Given, y-3 is the factor of

y^{3} + {2 y}^{2} - 9 y - 18

_____(1)
Thus y-3=0
=y=3
The value of y in equation 1 is substituted
=

{(3)}^{3} + 2 {(3)}^{3} - 9 (3) - 18

= 27+18-27-18
= 0
By using polynomial division, to decrease the third degree polynomial to a second degree polynomial. The formula for the polynomial division is:
Question Image

Question Image

Therefore the result of division is

y^{2} + 5 y + 6

Hence, we can write the polynomial as

y^{3} + {2 y}^{2} - 9 y - 18 = (y - 3) (y^{2} + 5 y + 6)

Solving the equation

y^{2} + 5 y + 6

y^{2} + 5 y + 6 = y^{2} + 2 y + 3 y + 6

=

y (y + 2) + 3 (y + 2)

=

(y + 2) (y + 3)

Hence, the correct answer is

(y + 2) (y + 3)

.

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