Class 10 (All in one) - PolynomialsContact Number: 9667591930 / 8527521718

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1.

For what value of *k*, 3 is a zero of the polynomial 2*x*^{2} + *x* + *k*?

2.

If 2 is a zero of polynomial *f*(*x*) = *ax*^{2} – 3(*a* – 1)*x* – 1, then find the value of *a*.

3.

If 2 and 3 are zeroes of polynomial 3*x*^{2} – 2*kx* + 2*m*, then find the values of *k* and *m*.

4.

What is the geometrical meaning of the zeroes of a polynomial?

5.

The graph of *y* = *p*(*x*) is given, where *p*(*x*) is a polynomial. Find the number of zeroes of *p*(*x*).

6.

Draw the graph of the linear polynomial *x* + 5 and also find the zeroes of the polynomial.

7.

Find the zeroes of the quadratic polynomial *y*^{2} – 92*y* + 1920.

8.

If zeroes α and β of a polynomial *x*^{2} – 7*x* + *k* are such that α – β = 1, then find the value of *k*.

9.

If α and β are the zeroes of the polynomial 2*y*^{2} + 7*y* + 5, then find the value of α + β + αβ.

10.

If α and β are the zeroes of the quadratic polynomial *f*(*x*) = 3*x*^{2} – 5*x* – 2, then evaluate α^{3} + β^{3}.

11.

If α and β are the zeroes of 4*x*^{2} + 3*x* + 7, then find the value of $\frac{1}{\alpha}+\frac{1}{\beta}.$

12.

If the sum and difference of zeroes of quadratic polynomial are –3 and –10, respectively. Then, find the difference of the squares of zeroes.

13.

If one of the zeroes of the cubic polynomial *x*^{3} + *ax*^{2} + *bx* + *c* is –1, then find the product of the other two zeroes.

14.

Two zeroes of cubic polynomial *ax*^{3} + 3*x*^{2} – *bx* –6 are –1 and –2. Find the third zero and values of *a* and *b*.

15.

Find the quadratic polynomial, whose sum of zeroes is 8 and their product is 12. Then, find the zeroes of the polynomial.

16.

Find the quadratic polynomial whose zeroes are $2\sqrt{\text{7}}$ and $-5\sqrt{\text{7}}.$

17.

Find the quadratic polynomial whose zeroes are 2 and –6, respectively. Verify the relation between the coefficients and zeroes of the polynomial.

18.

If 1 and –1 are zeroes of polynomial *Lx*^{4} + *Mx*^{3} + *Nx*^{2} + *Rx* + *P*, then show that *L* + *N* + *P* = *M* + *R*.

19.

How many polynomials will have their zeroes as –2 and 5?

20.

If α and β are zeroes of the quadratic polynomial *p*(*x*) = 6*x*^{2} + *x* – 1, then find the value of $\frac{\text{\alpha}}{\text{\beta}}+\frac{\text{\beta}}{\text{\alpha}}+2(\frac{1}{\text{\alpha}}+\frac{1}{\text{\beta}})+3\text{\alpha \beta}$

21.

If α and β are zeroes of the quadratic polynomial *f*(*x*) = *x*^{2} – 3*x* – 2, find a polynomial whose zeroes are

(i) $\frac{\text{2\alpha}}{\text{\beta}}\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}\frac{\text{2\beta}}{\text{\alpha}}$

(ii) $(\text{2\alpha}+\text{3\beta})\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}(\text{3\alpha}+\text{2\beta})$

(iii) $\frac{{\text{\alpha}}^{\text{2}}}{\text{\beta}}\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}\frac{{\text{\beta}}^{\text{2}}}{\text{\alpha}}$

(iv) $\frac{\text{1}}{\text{2\alpha}+\text{\beta}}\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}}\frac{\text{1}}{\text{2\beta}+\text{\alpha}}$

22.

On dividing a polynomial *p*(*x*) by 3*x* + 1, the quotient is 2*x* – 3 and the remainder is –2. Find *p*(*x*).

23.

What will be the quotient and the remainder on division of *ax*^{2} + *bx* + *c* by *px*^{3} + *qx*^{2} + *rx* + 5, *p* ≠ 0

24.

Divide the polynomial *p*(*x*) by the polynomial *g*(*x*) and verify the division algorithm in each of the following.

(i) *p*(*x*) = 2*x*^{4} – 2*x*^{3} – 5*x*^{2} – *x* + 8, *g*(*x*) = 2*x*^{2} + 4*x* + 3

(ii) *p*(*x*) = 10*x*^{4} + 17*x*^{3} – 62*x*^{2} + 30*x* – 3, *g*(*x*) = 2*x*^{2} + 7*x* – 1

25.

Find the value of *k*, for which polynomial *p*(*x*) is exactly divisible by polynomial *g*(*x*), in each of the following

(i) *p*(*x*) = *x*^{3} + 8*x*^{2} + *kx* + 18, *g*(*x*) = *x*^{2} + 6*x* + 9

(ii) *p*(*x*) = *x*^{4} + 10*x*^{3} + 25*x*^{2} + 15*x* + *k*, *g*(*x*) = 6*x* + 7

26.

If the polynomial 6*x*^{4} + 8*x*^{3} + 17*x*^{2} + 21*x* + 7 is divided by another polynomial 3*x*^{2} + 4*x* + 1, the remainder comes out to be *ax* + *b*, then find the values of *a* and *b*.

27.

A polynomial *g*(*x*) of degree zero is added to the polynomial 2*x*^{3} + 5*x*^{2} – 14*x* + 10, so that it becomes exactly divisible by 2*x* – 3. Find *g*(*x*).

28.

If the polynomial *f*(*x*) = 3*x*^{4} – 9*x*^{3} + *x*^{2} + 15*x* + *k* is completely divisible by 3*x*^{2} – 5, then find the value of *k* and hence the other two zeroes of the polynomial.

29.

The graphs of *y* = *p*(*x*), where *p*(*x*) is a polynomial in *x* are given. Find the number of zeroes of *p*(*x*) in each case. For each case, also state whether *p*(*x*) is linear or quadratic.

30.

Is the following statement True or False? Justify your answer. ‘If the zeroes of a quadratic polynomial *ax*^{2} + *bx* + *c* are both negative, then *a*, *b* and *c* all have the same sign.’

31.

If one zero of 2*x*^{2} – 3*x* + *k* is reciprocal to the other, then find the value of *k*.

32.

If sum of the squares of zeroes of the quadratic polynomial *f*(*x*) = *x*^{2} – 4*x* + *k* is 20, then find the value of *k*.

33.

If the zeroes of the quadratic polynomial *a**x*^{2} + *bx* + *c*, where *c* ≠ 0, are equal, then show that *c* and *a* have same sign.

34.

Can (*x* – l)be the remainder on division of a polynomial, *p*(*x*) by (2*x* + 3)? Justify your answer.

35.

Write whether the following expressions are polynomials or not. Give reasons for your answer.

(i) ${x}^{3}+\frac{1}{{x}^{2}}+\frac{1}{x}+1$

(ii) ${x}^{2}+x+3$

(iii) ${y}^{-1/2}-3y+2$

(iv) $\sqrt{2}{y}^{3}+\sqrt{3}y$

36.

If one zero of the polynomial (*a*^{2} + 9)*x*^{2} + 13*x* + 6*a* is reciprocal of the other, then find the value of *a*.

37.

The sum of remainders obtained when *x*^{3} + (*k* + 8) *x* + *k* is divided by *x* – 2 and when it is divided by *x* + 1, is 0. Find the value of *k*.

38.

Find the zeroes of the given polynomial by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials $7{y}^{2}-\frac{11}{3}y-\frac{2}{3}.$

39.

Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and their coefficients.

$q\left(x\right)=\sqrt{3}{x}^{2}+10x+7\sqrt{3}$

40.

Can the quadratic polynomial *x*^{2} + *kx* + *k* have equal zeroes for some odd integer *k* > 1?

41.

If the zeroes of the polynomial *ax*^{2} + *bx* + *b* = 0 are in the ratio *m* : *n*, then find the value of $\sqrt{\frac{m}{n}}+\sqrt{\frac{n}{m}}.$

42.

If α and β are the zeroes of the quadratic polynomial *f*(*x*) = *px*^{2} + *qx* + *r*, then evaluate $\frac{1}{p\text{\alpha}+q}+\frac{1}{p\text{\beta}+q}.$

43.

If α and β are the zeroes of the quadratic polynomial *p*(*s*) = 3*s*^{2} – 6*s* + 4, then find the value of $\frac{\text{\alpha}}{\text{\beta}}+\frac{\text{\beta}}{\text{\alpha}}+2(\frac{\text{1}}{\text{\alpha}}+\frac{\text{1}}{\text{\beta}})+3\text{\alpha \beta}.$

44.

If α and β are the zeroes of the quadratic polynomial *f*(*x*) = *x*^{2} – *px* + *q*, then prove that $\frac{{\text{\alpha}}^{\text{2}}}{{\text{\beta}}^{\text{2}}}+\frac{{\text{\beta}}^{\text{2}}}{{\text{\alpha}}^{\text{2}}}=\frac{{p}^{4}}{{q}^{2}}-\frac{4{p}^{2}}{q}+2.$

45.

Ajay, Ankit and Vijay respectively calculated the following polynomials with sum of the zeroes as 18 and product of the zeroes as 81.

*x*^{2} – 18*x* + 81, *x*^{2} + 18*x* – 81, 2*x*^{2} – 9*x* – 81

They discussed their solutions among themselves and point out mistakes in the calculations.

(i) Whose calculation is correct?

(ii) What are the values depict here?

46.

Remainder on dividing *x*^{3} + 2*x*^{3} + *kx* + 3 by *x* – 3 is 21. Ahmed was asked to find the quotient. He was little puzzled and was thinking how to proceed. His classmate Vidya helped him by suggesting that he should first find the value of *k* and then proceed further.

(i) Explain how the question was solved?

(ii) What value is indicated from Vidya's action?

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