Factor and reminder theorem Polynomial class 9 Practice Paper

CBSE MATH STUDY: Factor and reminder theorem :Polynomial class IX: 

Proof of this factor theorem Let  p ( x ) be a polynomial of degree greater than or equal to one and  a   be  areal number such that  p ( a...

Proof of this factor theorem

Let p(x) be a polynomial of degree greater than or equal to one and a be areal number such that p(a) = 0. Then, we have to show that (x – a) is a factor of p(x).

Let q(x) be the quotient when p(x) is divided by 

(x – a).

By remainder theorem,

Dividend = Divisor x Quotient + Remainder

p(x) = (x – a) x q(x) + p(a) [Remainder theorem]

⇒ p(x) = (x – a) x q(x) [p(a) = 0]

⇒ (x – a) is a factor of p(x)

Conversely, let (x – a) be a factor of p(x). Then we have to prove that p(a) = 0

Now,     (x – a) is a factor of p(x)

⇒ p(x), when divided by (x – a) gives remainder zero. 

But, by the remainder theorem, p(x) when divided by (x – a) gives the remainder equal to p(a).            ∴ p(a) = 0

Proof of remainder theorem.

Let q(x) be the quotient and r(x) be the remainder obtained when the polynomial p(x) is divided by (x–a).

Then, p(x) = (x–a) q(x) + r(x), where r(x) = 0 or some constant.

Let r(x) = c, where c is some constant. Then,

p(x) = (x–a) q(x) + c

Putting x = a in p(x) = (x–a) q(x) + c, we get

p(a) = (a–a) q(a) + c  ⇒ p(a) = 0 x q(a) + c  ⇒ p(a) = c

This shows that the remainder is p(a) when p(x) is divided by (x–a).

1. Factories

(i)   a2-b2- 4ac + 4c2                      (ii) 7x2 + 2 √14x + 2

(iii)4a² -4b²+4a+1                          (iv)  x⁴+y⁴-x²y²

(v) x⁶ - x³                                          

(vi)  x³-5x²-x+5

(vii)  x²+3√3x +6 

(viii)  a³(b-c)³+b³(c-a)³+c³(a-b)³

(ix)  .8a³-b³ -12a² b +6ab²        

(x) b.4x² +9y ²+ 25z² -12xy - 30yz +20xz

(xi)  x ³ + 4x ² + x – 6                (xii)  4x⁴ + 7x² – 2

(xii)  x ² - 2√3x – 45 

(xiii)  3 - 12(a - b)2

Q. Find degree of 5x³ -6x³y+10y²+11  [ans 4]  

Q. find the value of k if(x-2) is a factor of p(x)=k x^2 -- √2x +1    

[ And. (2√2- 1)/4 ]

Q. Find the remainder when x³+3x²+3x+1 when divided by 3x+1.

Q.  if x-1 and x-3 are the factors of p(x) x (raise to the power 3)-a x (raise to the power 2)-13x-b then find the value of a and b

Q. If(X2-1) is a factor of ax⁴+bx³+cx²+dx+e, show that  a + c + e = b + d =0

Q. prove that (x+y)³-(x-y)³-6y(x²-y²)=8y³

Ans:  x³ + 3x²y + 3xy² + y³ - (x³ - 3x ²y + 3xy² - y³) - 6yx² + 6y³ [(a + b)³ = a³ + 3a²b + 3ab² + b³

(a - b)³ = a³ - 3a²b + 3 ab² - b³]  = x³ + 3x²y + 3xy² + y³ - x³ + 3x²y - 3xy² + y³ - 6yx² + 6y³  

= 2y³ + 6 y³        = 8y³

Q. The polynomials {ax³-3x²+4} and {3x²-5x+a} when divided by {x-2} leave remainder "p" and "q" respectively. if p-2q+=a find the values of "a".   Ans, [a is –8.]

Q. If ( x − 4) is a factor of the polynomial 2 x ² + Ax + 12 and ( x − 5) is a factor of thepolynomial x ³ − 7 x ² + 11 x + B , then what is the value of ( A − 2 B )?

Q. If x -1/x =3; then find the value of x³ -1/x³                      [36]

Q. if a+b+c=7 and ab+bc+ca=20 find the value of (a+b+c)²

Q. The polynomial f(x)=x⁴-2x³+3x²-ax+b when divided by (x-1) and (x+1) leaves the remainders 5 and 19 respectively. Find the values of a and b. Hence, find the remainder when f(x) is divided by (x-2)

Q. check:  2x +1 is a factor of p(x)=4x³ + 4x² - x -1

Q. the degree of a polynomial A is 7 and that of the polynomial AB is 56, then find the degree of polynomial B

Ans: Degree of polynomial A = 7 and degree of polynomial AB = 56 ⇒ Degree ofpolynomial B = Degree of AB – Degree of A = 56 – 7 = 49

Q.The polynomial x⁴+bx³+59x²+cx+ 60 is exactly divisioble by x²+4x+3  find the values of b and c

Q.  If a+b+c=0 and a²+b²+c²=30. Then find (ab+bc+ca)

Q. Find the remainder when q(x)= x⁴ - 2x² +6x +3 is divided by x -2

Q. If a + b + c = 0 ( a² / bc ) + ( b²/ ca ) + ( c² / ab ) = 3

Q. If the remainder obtained on dividing the polynomial 2x ³ − 9x ² + 8x + 15 by (x − 1) is R ₁ and the remainder obtained on dividing the polynomial x ² − 10x + 50 by (x − 5) is R ₂, then what is the value of R ₁− R ₂?                [-9]

Q.  For what value of k the coefficient of x² in the polynominal 7x³ + 5x² - 2/k x² + 3 is - 6. [2/11]

Q.  Evaluate x⁴ + 1/x⁴ if x - 1/x = 6

Q. if (x-1)and(x+3) are the factors of polynomial  f(x) x³-px²-13x+q  , then what are the values of p and q?

Q. if both x-2 and x-1/2 are factors of px² +5x + r , show that p=r

Q. if the polynomial az³+4z²+3z-4 and z³-4z+a leave the same remainder when divided by z-3 find 'a'       [-1]

Q. Find the value of 64x³ + 125z³ , if 4x+5z = 19 and xz = 5.

Q.  if (2x-3) is a factor of 2x⁴-3x²+15x-15k find the value of (3k- √5 k)

Q. if a²+b²+c²-ab-bc-ca=0 then prove that a=b=c