IX Maths: Polynomials :Factor and reminder theorem

Proof of this factor theoremLet 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) + cPutting 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) 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 [4]
Q. find the value of k if(x-2) is a factor of p(x)=k x2 -- √2x +1 [ (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". [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 the polynomial x 3 − 7 x 2 + 11 x + B , then what is the value of ( A − 2 B )?
Q. f 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 of polynomial 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 -2Q. 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
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