What no should be added to 27x^3-54x^2+36 x -11. So that the resulting polynomial. Becomes divisible by (3x-2)
27x³ - 54x² + 36x - 11
3x - 2 =0
x=2/3
f(2/3)=27(2/3)³ - 54(2/3)² + 36(2/3) - 11 + y
0=27(8/27) - 54(4/9) + 36(2/3) - 11 + y
0=8 - 6(4) + 12(2) - 11 + y
0=-3 + y
3=y
Well, to make the polynomial divisible by (3x-2), we need to find a number that, when substituted for x, makes the polynomial equal to zero. So, let's substitute 2/3 (since 3x-2 = 0 when x = 2/3) into the polynomial:
27(2/3)^3 - 54(2/3)^2 + 36(2/3) - 11
27(8/27) - 54(4/9) + 24/3 - 11
8 - 24 + 8 - 11
-19
So, to make the polynomial divisible by (3x-2), we should add -19 to it. Why? Well, because then when we substitute x = 2/3 into the polynomial, it will equal zero. And trust me, zero is divisible by anything!
To find the remainder when dividing a polynomial by another polynomial, you can use the long division method. In this case, we want to find the value of "n" that should be added to the polynomial 27x^3 - 54x^2 + 36x - 11 so that the resulting polynomial is divisible by (3x - 2).
Step 1: Set up the long division:
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(3x - 2) | 27x^3 - 54x^2 + 36x - 11
Step 2: Divide the leading term of the polynomial being divided (27x^3) by the leading term of the divisor (3x). This gives us 9x^2.
Step 3: Multiply this term (9x^2) by the divisor (3x - 2) to get 27x^3 - 18x^2.
Step 4: Subtract this result (27x^3 - 18x^2) from the original polynomial (27x^3 - 54x^2 + 36x - 11) to get -36x^2 + 36x - 11.
Step 5: Repeat steps 2-4 with the new polynomial (-36x^2 + 36x - 11).
Step 6: Divide the leading term of the new polynomial (-36x^2) by the leading term of the divisor (3x). This gives us -12x.
Step 7: Multiply this term (-12x) by the divisor (3x - 2) to get -36x^2 + 24x.
Step 8: Subtract this result (-36x^2 + 24x) from the new polynomial (-36x^2 + 36x - 11) to get 12x - 11.
Step 9: Repeat steps 2-4 with the new polynomial (12x - 11).
Step 10: Divide the leading term of the new polynomial (12x) by the leading term of the divisor (3x). This gives us 4.
Step 11: Multiply this term (4) by the divisor (3x - 2) to get 12x - 8.
Step 12: Subtract this result (12x - 8) from the new polynomial (12x - 11) to get -19.
At this point, we have a remainder of -19.
For the resulting polynomial to be divisible by (3x - 2), the remainder should be zero. Therefore, we need to add 19 to the original polynomial:
27x^3 - 54x^2 + 36x - 11 + 19
Simplifying:
27x^3 - 54x^2 + 36x + 8
So the polynomial that becomes divisible by (3x - 2) when adding the number 19 is 27x^3 - 54x^2 + 36x + 8.