**NCERT Exemplar Problems Class 9 Maths – Polynomials**

**Question 1:**

Which one of the following is a polynomial?

**Solution:**

**Question 2:**

√2 is a polynomial of degree

(a) 2 (b) 0 (c) 1 (d)½

**Solution:**

(b) √2 = -√2x°. Hence, √2 is a polynomial of degree 0, because exponent of x is 0.

**Question 3:**

Degree of the polynomial 4x^{4} + Ox^{3} + Ox^{5} + 5x+ 7 is

(a) 4 (b) 5 (c) 3 (d) 7

**Solution:**

(a) Degree of 4x^{4} + Ox^{3} + Ox^{5} + 5x + 7 is equal to the highest power of variable x. Here, the highest power of x is 4,

Hence, the degree of a polynomial is 4.

**Question 4:**

Degree of the zero polynomial is

(a) 0 (b) 1

(c) any natural number (d) not defined

**Solution:**

(d) The degree of zero polynomial is not defined, because in zero polynomial, the coefficient of any variable is zero i.e., Ox^{2} or Ox^{5},etc.

Hence, we cannot exactly determine the degree of variable.

**Question 5:**

If p (x) = x2 – 2√2x + 1, then p (2√2) is equal to

(a) 0 (b) 1 (c) 4√2 (d) 8 √2 +1

**Solution:**

(b) Given, p(x) = x^{2} – 2√2x + 1 …(i)

On putting x = 2√2 in Eq. (i), we get

P(2√2) = (2√2)^{2}– (2√2)(2√2) + 1 =8-8+1=1

**Question 6:**

The value of the polynomial 5x – 4x^{2} + 3, when x = – 1 is

(a)-6 (b) 6 (c) 2 (d) -2

**Solution:**

(a) Let p (x) = 5x – 4x^{2} + 3 …(i)

On putting x = -1 in Eq. (i), we get

p(-1) = 5(-1) -4(-1)^{2} + 3= -5-4+3 = -6

**Question 7:**

If p (x) = x + 3, then p(x)+ p (- x) is equal to

(a) 3 (b) 2x (c) 0 (d) 6

**Solution:**

(d) Given p(x) = x+3, put x = -x in the given equation, we get p(-x) = -x+3

Now, p(x)+ p(-x) = x+ 3+ (-x)+ 3=6

**Question 8:**

Zero of the zero polynomial is

(a) 0 (b) 1 (c) any real number (d) not defined

**Solution:**

(c) Zero of the zero polynomial is any real number.

e.g., Let us consider zero polynomial be 0(x-k), where k is a real number For determining the zero, putx-k = 0=>x = k Hence, zero of the zero polynomial be any real number.

**Question 9:**

Zero of the polynomial p(x)=2x+5 is

(a) -2/5 (b) -5/2 (c)2/5 (d)5/2

**Solution:**

(b) Given, p(x) = 2x+5

For zero of the polynomial, put p(x) = 0 ∴ 2x + 5 = 0

=> -5/2

Hence, zero of the polynomial p(x) is -5/2.

**Question 10:**

One of the zeroes of the polynomial 2x^{2} + 7x – 4 is

(a) 2 (b)½ (c)-1 (d)-2

Thinking Process

(i) Firstly, determine the factor by using splitting method.

(ii) Further, put the factors equals to zero, then determine the values of x.

**Solution:**

(b) Let p (x) = 2x^{2} + 7x-4

= 2x^{2} + 8x-x-4 [by splitting middle term]

= 2x(x+ 4)-1(x+ 4)

= (2x-1)(x+ 4)

For zeroes of p(x), put p(x) = 0=> (2x -1) (x + 4) = 0

=> 2x-1 = 0 and x+4 = 0

=> x = ½ and x = -4

Hence, one of the zeroes of the polynomial p(x) is ½.

**Question 11:**

If x^{51} + 51 is divided by x + 1, then the remainder is

(a) 0 (b) 1 (c) 49 (d) 50

**Solution:**

(d) Let p(x) = x^{51} + 51 . …(i)

When we divide p(x) by x+1, we get the remainder p(-1)

On putting x= -1 in Eq. (i), we get p(-1) = (-1)^{51} + 51

= -1 + 51 = 50

Hence, the remainder is 50.

**Question 12:**

If x + 1 is a factor of the polynomial 2x^{2} + kx, then the value of k is

(a) -3 (b) 4 (c) 2 (d)-2

**Solution:**

(c) Let p(x) = 2x^{2} + kx

Since, (x + 1) is a factor of p(x), then

p(-1)=0

2(-1)2 + k(-1) = 0

=> 2-k = 0 => k= 2

Hence, the value of k is 2.

**Question 13:**

x + 1 is a factor of the polynomial

(a) x^{3} + x^{2} – x +1 (b) x^{3} + x^{2} + x+1

(c) x^{4} + x^{3} + x^{2} +1 (d) x^{4} + 3x^{3} + 3x^{2} + x +1

**Solution:**

(b) Let assume (x + 1) is a factor of x^{3} + x^{2} + x+1.

So, x = -1 is zero of x^{3} + x^{2} + x+1

(-1)^{3} + (-1)^{2} + (-1) + 1 = 0 => -1+1-1 + 1 = 0 => 0 = 0 Hence, our assumption is true.

**Question 14:**

One of the factors of (25x^{2} – 1) + (1 + 5x)^{2} is

(a) 5 + x (b) 5 – x (c) 5x -1 (d) 10x

**Solution:**

(d) Now, (25x^{2} -1) + (1 + 5x)^{2}

= 25x^{2} -1 + 1 + 25x^{2} + 10x [using identity, (a + b)^{2} = a^{2} + b^{2} + 2ab]

= 50x^{2} + 10x = 10x (5x+ 1)

Hence, one of the factor of given polynomial is 10x.

**Question 15:**

The value of 249^{2} – 248^{2} is

(a) 1^{2} (b) 477 (c) 487 (d) 497

**Solution:**

(d) Now, 249^{2} – 248^{2} = (249 + 248) (249 – 248) [using identity, a^{2} – b^{2} = (a – b)(a + b)]

= 497 x 1 = 497.

**Question 16:**

The factorization of 4x^{2} + 8x+ 3 is

(a) (x +1) (x + 3) (b) (2x+1) (2x + 3)

(c) (2x + 2) (2x + 5) (d) (2x -1) (2x – 3)

**Solution:**

(b) Now, 4x^{2} + 8x + 3= 4x^{2} + 6x + 2x + 3 [by splitting middle term]

= 2x(2x+ 3) + 1 (2x+ 3)

= (2x + 3) (2x + 1)

**Question 17:**

Which of the following is a factor of (x+ y)^{3} – (x^{3} + y^{3})?

(a) x^{2} + y^{2} + 2 xy (b) x^{2} + y^{2} – xy (c) xy^{2} (d) 3xy

**Solution:**

(d) Now, (x+ y)3 – (x^{3} + y^{3}) = (x + y) – (x + y)(x^{2}– xy + y^{2})

[using identity, a^{3} + b^{3} = (a + b)(a^{2} -ab+ b^{2})] = (x+ y)[(x+ y)^{2} -(x^{2} -xy+ y^{2})]

= (x+ y)(x^{2}+ y^{2}+ 2xy- x^{2}+ xy- y^{2})

[using identity, (a + b)^{2} = a^{2} + b^{2} + 2 ab)]

= (x + y) (3xy)

Hence, one of the factor of given polynomial is 3xy.

**Question 18:**

The coefficient of x in the expansion of (x + 3)^{3} is

(a) 1 (b) 9 (c) 18 (d) 27

**Solution:**

(d) Now, (x + 3)^{3} = x^{3} + 3^{3} + 3x (3)(x + 3)

[using identity, (a + b)^{3} = a^{3} + b^{3} + 3ab (a + b)]

= x^{3} + 27 + 9x (x + 3)

= x^{3} +27 + 9x^{2}+27x Hence, the coefficient of x in (x + 3)^{3} is 27.

**Question 19:**

**Solution:**

**Question 20:**

**Solution:**

**Question 21:**

If a + b + c =0, then a^{3}+b^{3} + c^{3} is equal to

(a) 0 (b) abc

(c) 3abc (d) 2abc

**Solution:**

(d) Now, a^{3}+b^{3} + c^{3}= (a+ b + c) (a^{2} + b^{2} + c^{2} – ab – be – ca) + 3abc

[using identity, a^{3}+b^{3} + c^{3} – 3 abc = (a + b + c)(a^{2} + b^{2} + c^{2} – ab – be -ca)] = 0 + 3abc [∴ a + b + c = 0, given]

a^{3}+b^{3} + c^{3} = 3abc

**Exercise 2.2: Short Answer Type Questions**

**Question 1:**

Which of the following expressions are polynomials? Justify your answer,

**Solution:**

**Question 2:**

Write whether the following statements are true or false. Justify your answer. ’

(i) A Binomial can have atmost two terms.

(ii) Every polynomial is a Binomial.

(iii) A binomial ipay have degree 5.

(iv) Zero of a polynomial is always 0.

(v) A polynomial cannot have more than one zero.

(vi) The degree of the sum of two polynomials each of degree 5 is always 5.

**Solution:**

(i) False, because a binomial has exactly two terms.

(ii) False, because every polynomial is not a binomial .

e.g., (a) 3x^{2} + 4x + 5 [polynomial but hot a binomial]

(b) 3x^{2} + 5 [polynomial and also a binomial]

(Hi) True, because a binomial is a polynomial whose degree is a whole number greater than equal to one. So, it may have degree 5.

(iv) False, because zero of a polynomial can be any real number e.g., p(x) = x – 2, then 2 is a zero of polynomial p(x).

(v) False, because a polynomial can have any number of zeroes. It depends upon the degree of the polynomial

e.g., p(x) = x^{2} -2, as degree pf p(x) is 2 ,so it has two degree, so it has two zeroes i.e., √2 and —√2.

(vi) False, because the sum of any two polynomials of same degree is not always same degree.

e.g., Let f(x) = x^{4} + 2 and g(x) = -x^{4} + 4x^{3} + 2x

∴ Sum of two polynomials,

f{x) + g(x) = x^{4} + 2 + (-x^{4} + 4x^{3} + 2x)

= 4x^{3} + 2x + 2 which is not a polynomial of degree 4.

**Exercise 2.3: Short Answer Type Questions**

**Question 1:**

Classify the following polynomials as polynomials in one variable, two variables etc.

(i) x^{2} + x +1 (ii) y^{3} – 5y

(iii) xy+ yz +zx (iv) x^{2} – Zxy + y^{2} +1

**Solution:**

(i) Polynomial x^{2}+ x+ 1 is a one variable polynomial, because it contains only one variable i.e., x.

(ii) Polynomial y^{3} – 5y is a one variable polynomial, because it contains only one variable i.e., y.

(iii) Polynomial xy+ yz+ zx is a three variables polynomial, because it contains three variables x, y and z.

(iv) Polynomial x^{2} – Zxy + y^{2} +1 is a two variables pplynomial, because it contains two variables x and y.

**Question 2:**

Determine the degree of each of the following polynomials.

(i) 2x-1 (ii) -10

(iii) x^{3}-9x+ 3x^{5} (iv) y^{3}(1-y^{4})

**Solution:**

(i) Degree of polynomial 2x-1 is one, Decause the maximum exponent of x is one.

(ii) Degree of polynomial -10 or -10x° is zero, because the exponent of x is zero.

(iii) Degree of polynomial x3 – 9x + 3xs is five, because the maximum exponent of x is five.

(iv) Degree of polynomial y^{3}(1-y^{4}) or y^{3} – y^{7} is seven, because the maximum exponent of y is seven.

**Question 3:**

**Solution:**

**Question 4:**

Write the coefficient of x^{2} in each of the following

**Solution:**

**Question 5:**

Classify the following as a constant, linear, quadratic and cubic polynomials

Thinking Process

(i) Firstly check the maximum exponent of the variable..

(ii) If the maximum exponent of a variable is 0, then it is a constant polynomial.

(iii) If the maximum exponent of a variable is 1, then it is a linear polynomial.

(iv) If the maximum exponent of a variable is 2, then it is a quadratic polynomial.

(v) If the maximum exponent of a variable is 3, then it is a cubic polynomial.

**Solution:**

(i) Polynomial 2 – x^{2} + x^{3} is a cubic polynomial, because maximum exponent of x is 3.

(ii) Polynomial 3x^{3} is a cublic polynomial, because maximum exponent of x is 3.

(iii) Polynomial 5t -√7 is a linear polynomial, because maximum exponent of t is 1.

(iv) Polynomial 4- 5y^{2} is a quadratic polynomial, because maximum exponent of y is 2.

(v) Polynomial 3 is a constant polynomial, because the exponent of variable is 0. ’

(vi) Polynomial 2 + x is a linear polynomial, because maximum exponent of x is 1.

(vii) Polynomial y^{3} – y is a cubic polynomial, because maximum exponent of y is 3.

(viii) Polynomial 1 + x+ x^{2} is a quadratic polynomial, because maximum exponent of xis 2.

(ix) Polynomial t^{2} is a quadratic polynomial, because maximum exponent of t is 2.

(x) Polynomial √2x-1 is a linear polynomial, because maximum exponent of xis 1.

**Question 6:**

Give an example of a polynomial, which is

(i) monomial of degree 1.

(ii) -binomial of degree 20.

(iii) trinomial of degree 2.

**Solution:**

(i) The example of monomial of degree 1 is 5y or 10x.

(ii) The example of binomial of degree 20 is 6x^{20} + x^{11} or x^{20} +1

(iii) The example of trinomial of degree 2 is x^{2} – 5x+ 4 or 2x^{2} -x-1

**Question 7:**

Find the value of the polynomial 3x^{3} – 4x^{2} + 7x – 5, when x = 3 and also when x = -3.

**Solution:**

Let p(x) =3x^{3} – 4x^{2} + 7x – 5

At x= 3, p(3) = 3(3)^{3} – 4(3)^{2} + 7(3) – 5

= 3×27-4×9 + 21-5 = 81-36+21-5 P( 3) =61

At x = -3, p(-3)= 3(—3)^{3} – 4(-3)^{2} + 7(-3)- 5

= 3(-27)-4×9-21-5 = -81-36-21-5 = -143 p(-3) = -143

Hence, the value of the given polynomial at x = 3 and x = -3 are 61 and -143, respectively.

**Question 8:**

If p (x) = x^{2} – 4x + 3, then evaluate p(2) – p (-1) + p ( ½).

Solution:

**Question 9:**

Find p(0), p( 1) and p(-2) for the following polynomials

(i) p(x) = 10x – 4x^{2} – 3 (ii) p(y) = (y + 2)(y – 2)

**Solution:**

(i) Given, polynomial is

p(x) = 10x – 4x^{2} – 3

On putting x = 0,1 and – 2, respectively in Eq. (i), we get p(0) = 10(0)-4(0)^{2} -3 = 0-0-3= -3

p(1) = 10 (1) — 4 (1 )^{2} -3

= 10-4-3= 10-7= 3

and p(-2) =10 (-2)- 4 (-2)^{2} – 3

= -20-4×4-3 =-20-16-3=-39

Hence, the values of p(0), p(1) and p(-2) are respectively, -3,3 and – 39.

(ii) Given, polynomial isp(y) = (y+2)(y-2)

On putting y =0,1 and -2, respectively in Eq. (i), we get p(0) =(0+2)(0-2)= -4

p(1) = (1 + 2)(1-2)

= 3 x (-1) = -3

and p(-2) = (-2 + 2)(-2 -2)

=0 (-4) = 0

Hence, the values of p(0),p(1) and p(-2) are respectively,-4,-3 and 0.

**Question 10:**

Verify whether the following are true or false.

(i) -3 is a zero of at – 3

(ii) -1/3 is a zero of 3x+1

(iii) -4/5 is a zero of 4 – 5y

(iv) 0 and 2 are the zeroes of t^{2} – 2t

(v) -3 is a zero of y^{2} +y-6

**Solution:**

**Question 11:**

Find the zeroes of the polynomial in each of the following,

(i) p(x)=x – 4 (ii) g(x)= 3 – 6x

(iii) q(x) = 2x – 7 (iv) h(y) = 2y

**Solution:**

(i) Given, polynomial is

p(x) = x- 4

For zero of polynomial, put p(x) = x-4 = 0

=> x= 4

Hence, zero of polynomial is 4.

(ii) Given, polynomial is

g(x) = 3-6x

For zero of polynomial, put g(x) = 0

3-6x= 0 => 6x =3 => x=1/2.

Hence, zero of polynomial is X

(iii) Given, polynomial is q(x) = 2x -7 For zero of polynomial, put q(x) = 2x-7 = 0

2x= 7 => x =7/2

Hence, zero of polynomial is

(iv) Given polynomial h(y) = 2 y For zero of polynomial, put h(y) = 0

2y= 0

Hence, the zero of polynomial is 0,

**Question 12:**

Find the zeroes of the polynomial p(x)= (x – 2)^{2} – (x+ 2)^{2}.

**Solution:**

Given, polynomial is p(x) = (x – 2)^{2} – (x+ 2)^{2}

For zeroes of polynomial, put p(x) = 0

(x – 2)^{2} – (x+ 2)^{2}=0

(x-2 + x+2)(x-2-x-2) = 0 [using identity, a^{2}-b^{2} =(a-b)(a + b)] =» (2x)(-4) = 0

**Question 13:**

By actual division, find the quotient and the remainder when the first

polynomial is divided by the second polynomial x^{4} + 1 and x-1.

**Solution:**

**Question 14:**

By remainder theorem, find the remainder when p(x) is divided by g(x)

(i) p(x) = x^{3}-2x^{2}-4x-1, g(x)=x + 1

(ii) p(x) = x^{3} -3x^{2} + 4x + 50, g(x)= x – 3

(iii) p(x) = x^{3} – 12x^{2} + 14x -3, g(x)= 2x – 1 – 1

(iv) ‘p(x) = x^{3}-6x^{2}+2x-4, g(x) = 1 -(3/2) x

**Solution:**

**Question 15:**

Check whether p(x) is a multiple of g(x) or not

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

(ii) p(x) = 2x^{3} – 11x^{2} – 4x+ 5, g(x) = 2x + l

Thinking Process

(i) Firstly, find the zero of g(x) and then put the value of in p(x) and simplify it.

(ii) lf p(a) = Q then p(x) is a multiple of g(x) and f(p(a) # Q then p(x) is not a multiple of g(x) where ‘a’ is a zero of g (x)).

**Solution:**

**Question 16:**

Show that,

(i) x + 3 is a factor of 69 + 11c – x^{2} + x^{3}

(ii) 2x – 3 is a factor of x + 2x^{3} -9x^{2} +12

**Solution:**

**Question 17:**

Determine which of the following polynomial has x – 2 a factor (i) 3x^{2} + 6x – 24 (ii) 4x^{2}+ x – 2

**Solution:**

**Question 18:**

Show that p-1 is a factor of p^{10} -1 and also of p^{11} -1.

**Solution:**

Let g (p) = p^{10} -1

and h(p) = p^{11} -1.

On putting p=1 in Eq. (i), we get

g(1)=1^{10}-1= 1-1=0 Hence, p-1 is a factor of g(p).

Again, putting p = 1 in Eq. (ii), we get

h (1) = (1)^{11} —1 = 1 —1 = 0 Hence, p -1 is a factor of h(p).

**Question 19:**

For what value of m is x^{3} -2mx^{2} +16 divisible by x + 2?

**Solution:**

Let p(x) = x^{3} -2mx^{2} +16

Since, p(x) is divisible by (x+2), then remainder = 0

P(-2) = 0

=> (-2)^{3} -2m(-2)^{2} + 16=0

=> -8-8m+16=0

=> 8=8 m

m = 1

Hence, the value of m is 1 .

**Question 20:**

If x + 2a is a factor of a^{5} -4a^{2}x^{3} +2x + 2a +3, then find the value of a.

**Solution:**

Let p(x) =a^{5} -4a^{2}x^{3} +2x + 2a +3

Since, x + 2a is a factor of p(x), then put p(-2a) = 0 (-2a)^{5 }– 4a^{2} (-2a)^{3} + 2(-2a) + 2a + 3 = 0 => -32a^{5} + 32a^{5} -4a + 2a+ 3 = 0

=> -2a + 3=0

2a =3

a = 3/2.

Hence, the value of a is 3/2.

**Question 21:**

Find the value of m, so that 2x -1 be a factor of

8x^{4} +4x^{3} -16x^{2} +10x+07.

**Solution:**

**Question 22:**

If x +1 is a factor of ax^{3} +x^{2} -2x+4a-9, then find the value of a.

**Solution:**

**Question 23:**

Factorise

(i) x^{2} + 9x +18 (ii) 6x^{2} +7x -3

(iii) 2x^{2} -7x.-15 (iv) 84-2r-2r^{2}

**Solution:**

**Question 24:**

Factorise

(i) 2x^{3} -3x^{2} -17x + 30 (ii) x^{3} -6x^{2} +11 x-6

(iii) x^{3} + x^{2} – 4x – 4 (iv) 3x^{3} – x^{2} – 3x +1

Thinking Process

(i) Firstly, find the prime factors of constant term in given polynomial.

(ii) Substitute these factors in place of x in given polynomial and find the minimum factor which satisfies given polynomial and write it in the form of a linear polynomial.

(iii) Now, adjust the given polynomial in such a way that it becomes the product of two factors, one of them is a linear polynomial and other is a quadratic polynomial.

(iv) Further, determine the factor of quadratic polynomial by splitting the middle term.

**Solution:**

**Question 25:**

Using suitable identity, evaluate the following

(i) 103^{3} (ii) 101 x 102 (iii) 999^{2}

Thinking Process

Firstlyadjust the given number into two number such that one is a multiple of 10 and use the proper identity.

**Solution:**

**Question 26:**

Factorise the following

**Solution:**

**Question 27:**

Factorise the following

(i) 9x^{2} -12x+ 3 (ii) 9x^{2} -12xy + 4

**Solution:**

**Question 28:**

Expand the following

(i) (4a-b + 2c)^{2} (ii) (3a – 5b – c)^{2}

(iii) (-x + 2y-3z)^{2}

**Solution:**

**Question 29:**

Factorise the following

(i) 9x^{2} +4y^{2} + 16z^{2} +12xy-16yz -24xz

(ii) 25x^{2} + 16y^{2} + 4z^{2} – 40xy + 16yz – 20xz

iii) 16x^{2} + 4y^{2} + 9z^{2} – 16xy – 12yz + 24xz

**Solution:**

**Question 30:**

If a + b + c = 9 and ab + bc + ca = 26, find a^{2} + b^{2} +c^{2}.

**Solution:**

**Question 31:**

Expand the following

**Solution:**

**Question 32:**

Factorise the following

**Solution:**

**Question 33:**

Find the following products

**Solution:**

**Question 34:**

Factorise

(i) 1 + 64x^{3 }(ii) a^{3} -2√2b^{3}

Thinking Process

(i) Firstly, adjust the given number either in the farm 0f a^{3} + b^{3} or a^{3} -b^{3}

(ii) Further, use any of the identities i.e., a^{3} + b^{3} =(a + b)(a^{2}+b^{2}-ab) and a^{3} -b^{3} =(a-b)(a^{2} + b^{2} + ab), then simplify it, to get the factor.

**Solution:**

**Question 35:**

Fi nd (2x – y + 3z) (4x^{2} + y^{2} + 9z^{2 }+ 2xy + 3yz – 6xz).

**Solution:**

**Question 36:**

Factorise

(i) a^{3} -8b^{3} -64c^{3} -2Aabc

(ii) 2√2a^{3} +8b^{3} -27c^{3} +18√2abc

**Solution:**

**Question 37:**

Without actually calculating the cubes, find the value of 36xy-36xy = 0

Thinking Process

In this method firstly check the values of a + b+ c, then . if a + b+c = Q, now use the identity a^{3} + b^{3} + c^{3} = 3abc.

**Solution:**

**Question 38:**

Without finding the cubes, factorise (x- 2y)^{3} + (2y – 3z)^{3} + (3z – x)^{3}.

**Solution:**

We know that,

a^{3} + b^{3} + c^{3} – 3 abc = (a + b + c)(a^{2} + b^{2} + c^{2} -ab-bc-ca)

Also, if a + b + c = 0, then a^{3} + b^{3} + c^{3} = 3abc

Here, we see that (x-2y) +(2y-3z)+ (3z-x) = 0

Therefore, (x-2y)3 + (2y-3z)3 + (3z-x)3 = 3(x-2y)(2y-3z)(3z-x).

**Question 39:**

Find the value of

(i) x^{3} +y^{3} -12xy + 64,when x+y = -4.

(ii) x^{3} -8y^{3} -36xy-216,when x = 2y + 6.

**Solution:**

**Question 40:**

Give possible expression for the length and breadth of the rectangle whose area is given by 4a^{2} +4a – 3.

**Solution:**

Given, area of rectangle = 4a^{2} + 6a-2a-3

= 4a^{2} + 4a – 3 [by splitting middle term]

= 2a(2a + 3) -1 (2a + 3) = (2a – 1)(2a + 3) Hence, possible length = 2a -1 and breadth = 2a + 3

**Exercise 2.4: Long Answer Type Questions**

**Question 1:**

If the polynomials az^{3} +4z^{2} + 3z-4 and z^{3}-4z + o leave the same remainder when divided by z – 3, find the value of a.

**Solution:**

Let p_{1}(z) = az^{3} +4z^{2} + 3z-4 and p_{2}(z) = z^{3}-4z + o

When we divide p_{1}(z) by z – 3, then we get the remainder p,(3).

Now, p_{1}(3) = a(3)3 + 4(3)2 + 3(3) – 4

= 27a+ 36+ 9-4= 27a+ 41 When we divide p_{2}(z) by z-3 then we get the remainder p_{2}(3).

Now, p_{2}(3) = (3)^{3}-4(3)+a

= 27-12 + a = 15+a According to’ the question, both the remainders are same.

p_{1}(3)= p_{2}(3)

27a+41 = 15+a

27a-a = 15-41 .

26a = 26 a = -1

**Question 2:**

The polynomial p{x) = x^{4} -2x^{3} + 3x^{2} -ax+3a-7 when divided by x+1 leaves the remainder 19. Find the values of a. Also, find the remainder when p(x) is divided by x+ 2.

**Solution:**

**Question 3:**

If both x – 2 and x -(1/2) are factors of px^{2}+ 5x+r, then show that p = r.

**Solution:**

**Question 4:**

Without actual division, prove that 2x^{4} – 5x^{3} + 2x^{2} – x+ 2 is divisible by x^{2}-3x+2

Thinking Process

(i) firstly, determine the factors of quadratic polynomial by splitting middle term.

(ii) The two different values of zeroes put in biquadratic polynomial.

(iii) In both the case if remainder is zero, then biquadratic polynomial is divisible by

quadratic polynomial.

**Solution:**

Let p(x) = 2x^{4} – 5x^{3} + 2x^{2} – x+ 2 firstly, factorise x^{2}-3x+2.

Now, x^{2}-3x+2 = x^{2}-2x-x+2 [by splitting middle term]

= x(x-2)-1 (x-2)= (x-1)(x-2)

Hence, 0 of x^{2}-3x+2 are land 2.

We have to prove that, 2x^{4} – 5x^{3} + 2x^{2} – x+ 2 is divisible by x^{2}-3x+2 i.e., to prove that p (1) =0 and p(2) =0

Now, p(1) = 2(1)^{4} – 5(1)^{3} + 2(1)^{2} -1 + 2 =2-5+2-1+2=6-6=0

and p(2) = 2(2)^{4} – 5(2)^{3} + 2(2)^{2} – 2 + 2 = 2x^{16}-5x^{8}+2x^{4}+ 0 = 32 – 40 + 8 = 40 – 40 =0

Hence, p(x) is divisible by x^{2}-3x+2.

**Question 5:**

Simplify (2x- 5y)^{3} – (2x+ 5y)^{3}.

**Solution:**

(2x -5y)^{3} – (2x + 5y)^{3} = [(2x)^{3} – (5y)^{3} – 3(2x)(5y)(2x – 5y)]

-[(2x)^{3} + (5y)^{3} + 3(2x)(5y)(2x+5y)]

[using identity, (a – b)^{3} = a^{3} -b^{3} – 3ab and (a + b)^{3} =a^{3} +b^{3} + 3ab ] = (2x)^{3} – (5y)^{3} – 30xy(2x – 5y) – (2x)^{3} – (5y)^{3} – 30xy (2x + 5y)

= -2 (5y)^{3} – 30xy(2x – 5y + 2x + 5y)

= -2 x 125y^{3} – 30xy(4x) = -250y^{3} -120x^{2}y

**Question 6:**

Multiply x^{2} + 4y^{2} + z^{2} + 2xy + xz – 2yz by (-z + x-2y).

**Solution:**

**Question 7:**

**Solution:**

**Question 8:**

If a+b+c= 5 and ab+bc+ca =10, then prove that a^{3} +b^{3} +c^{3 }– 3abc = -25.

**Solution:**

**Question 9:**

Prove that (a +b +c)^{3} -a^{3} -b^{3 }– c^{3} =3(a +b)(b +c)(c +a).

**Solution:**