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1.

Factorize `9z^3-27 z^2-100 z+300 ,`if it is given that `(3z+10)`is a factor of it.

Answer» `9z^3-27z^2-100z+300=(3z+10)(Az^2+bz+c)`
`=3AZ^3+3BZ^2+10AZ^2+3CZ+10NZ+10C`
`=3AZ^3+Z^2(3B+10A)+Z(3C+10B)+10C`
`3A=9`
`A=3`
`3B+10A=-27`
`B=-19`
`3C+10B=-100`
`10C=300`
`C=30`
`(3Z+10)(3Z^2-19Z+30)=0`
`3Z^2-9z-10z+30=0`
`3z(z-3)-10(z-3)=0`
`(3z-10)(z-3)=0`
`(3z+10)(3z-10)(z-3)`.
Discussion
2.

Use factor theorem to verify that `x+a`is a factor of `x^n+a^n`for any odd positive integer.

Answer» `(-1)*(-1)=`
`1*(-1)=(-1)`
`(-1)*(-1)*(-1)=(-1)^3=(-1)`
`(-1)^n=1` when n is even
`(-1)^n=-1` when n is odd.
`f(x)=x^n+a^n`
`x=-a`and then Remainder=0
`f(-a)=(-a)^n+a^n`
`=(-a)^n(a)^n+a^n`
`=a^n[1-(-1)^n]=0`
`1+(-1)^n=0`
`(-1)^n=-1`
It is possible only when n is odd.
Discussion
3.

Write the degrees of each of the followingpolynomials:`7x^3+4x^2-3x+12`(ii) `12-x+2x^3`(iii) `5y-sqrt(2)`(iv) 7 (v) 0

Answer» degree= maximum power of variable
1) 3 degree
2) 3 degree
3) 1 degree
4) 0 degree
5) degree is not defined.
Discussion
4.

If `x=4/3`is a root of the polynomial `f(x)=6x^3-11 x^2+k x-20 ,`find the value of `k`

Answer» x=a is root of f(x)
f(a)=0
f(4/3)=0
`6*(4/3)^3-11(4/3)^2+k(4/3)-20=0`
`6*64/27-11*16/9+4/3k-20=0`
`(384-176*3+9*4k-20*27)/27=0`
`384-528+36k-540=0`
`36k=1068-384`
`k=19`.
Discussion
5.

What must be added to `x^4+2x^3-2x^2+x-1`so that the result is exactly divisible by `x^2+2x-3.`

Answer» Dividend=divisor*quotient+remainder
p-(-x+2)=(divisor*quotient)
p+(x-2)=D*Q
(x-2)
2 should be added .
Discussion
6.

Find the remainder when `f(x)=x^3-6x^2+2x-4`is divided by `g(x)=3x-1.`

Answer» When f(x) is divided by then remainder =f(a)
`g(x)=3x-1=3(x-1/3)`
`a=1/3`
Rem`=f(1/3)=(1/3)^3-6(1/3)^2+2(1/3)-4`
`=1/27-6/9+2/3-4`
`=1/27-4`
`=-107/27`.
Discussion
7.

If `x-2`is a factor of each of the following twopolynomials, find the values of `a`in each case.`x^3-2a x^2+a x-1``x^5-3x^4-a x^3+3a x^2+2a x+4`

Answer» (x-a) is factor of f(x)
f(a)=0
`f(x)=x^3-2ax^2+ax-1`
`f(2)=0`
`2^3-2a2^2+a*2-1=0`
`8-8a+2a-1=0`
`a=7/6`
`g(x)=x^5-3x^4-ax^3+2ax+4`
`g(2)=0`
`32-48-8a-12a+4a+4=0`
`-16a-12=0`
`a=-3/4`.
Discussion
8.

What must be added to `3x^3+x^2-22 x+9`so that the result is exactly divisible by `3x^2+7x-6?`

Answer» `3x^3+x^2-22x+9=x(3x^2+7x-6)-2(3x^2+7x-6)-2x-3`
`R=-2x-3`
We need to add 2x+3.
Discussion
9.

If the polynomials `a x^3+3x^2-13`and `2x^3-5x+a ,`when divided by `(x-2)`leave the same remainder, find the value of `a`.

Answer» `f(x)=ax^3+3x^2-13`
`g(x)=2x^3-5x+9`
`x-2=0`
`x=2`
`a2^3+3*2^2-13=2*2^3-5*2+9`
`8a+12-13=16-10+9`
`7a=6+1`
`7a=7`
`a=1`.
Discussion
10.

What must be subtracted from `4x^4-2x^3-6x^2+x-5`so that the result is exactly divisible by `2x^2+x-1`

Answer» -6 should be subtracted, so that remainder will be 0.
Discussion