10th Maths 2.3
NCERT Class 10th solution of Exercise 2.2
NCERT Class 10th solution of Exercise 2.1
Exercise 2.3
Q1. Divide the polynomial `p(x)` by the polynomial `g(x)` and find the quotient and remainder in each of the following :
i) `p(x)=x^3-3x^2+5x-3, g(x)=x^2-2`
ii) `p(x)=x^4-3x^2+4x+5, g(x)=x^2+1-x`
Q2. Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial :
i) `t^2-3, 2t^4+3t^3-2t^2-9t-12`
ii) `x^2+3x+1, 3x^4+5x^3-7x^2+2x+2`
iii) `x^3-3x+1, x^5-4x^3+x^2+3x+1`
Q3. Obtain all other zeroes of `3x^4+6x^3-2x^2-10x-5`, if two of its zeroes are `sqrt(5/3)` and `-sqrt(5/3)`.
Sol.
`sqrt(5/3)` and `-sqrt(5/3)` are two zeroes of the given polynomial. So,
`=(x + sqrt(5/3))(x - sqrt(5/3))`
`=(x^2 - 5/3)`
`=(3x^2 - 5)`
Q4. On dividing `x^3-3x^2+x+2` by a polynomial `g(x)`, the quotient and remainder were `x-2` and `-2x+4`, respectively. Find `g(x)`.
Sol.
By using Euclid's Division Algorithm
`g(x)(x-2)+(-2x+4)=x^3-3x^2+x+2`
`g(x)(x-2)=x^3-3x^2+x+2+2x-4`
`g(x)=(x^3-3x^2+3x-2)/(x-2)`
Q5. Give examples of polynomials `p(x),g(x),q(x)` and `r(x)` which satisfy the division algorithm and i) deg`p(x)`=deg`q(x)` ii) deg `q(x)`=deg`r(x)` iii) deg `r(x)=0`.
Answer
i) `p(x)=2x^2-2x+14, g(x)=2,`
`q(x)=x^2-x+7, r(x)=0`
ii) `p(x)=x^3+x^2+x+1, g(x)=x^2-1,`
`q(x)=x+1, r(x)=2x + 2`
iii) `p(x) = x^3 + 2x^2 - x + 2, g(x)=(x^2-1), ``q(x)=x+2, r(x)=4`
There can be several example in each of (i), (ii), and (iii).
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