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) = 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).

Check Your understanding

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 5x^{3 -}6x³y+10y²+11                        [4]

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