1 Crore+ students have signed up on EduRev. Have you? 
If the polynomial f(x) = x^{2 }+ kx  15, is exactly divisible by x  5, then the value of k is _______
x^{2 }+ kx  15 is exactly divisible by x  5
Dividing, x^{2 }+ kx + 15 by x  5
We get, 5k + 10 as remainder.
Since, x^{2 }+ kx  15 is exactly divisible by 2x  5
∴ 5k + 10 = 0
k = 2
The real number that should be subtracted from the polynomial f(x) = 15x^{5 }+ 70x^{4 }+ 35x^{3 } 135x^{2 } 40x  11 so that it is exactly divisible by 5x^{4 }+ 10x^{3 } 15x^{2 } 5x is ____________
On dividing, 15x^{5 }+ 70x^{4 }+ 35x^{3 } 135x^{2 } 40x  11 by 5x^{4 }+ 10x^{3 } 15x^{2 } 5x
We get, 3x + 8 as quotient and remainder as 11.
So if we subtract 11 from 15x^{5 }+ 70x^{4 }+ 35x^{3 } 135x^{2 } 40x  11 it will be exactly divisible by 5x^{4 }+ 10x^{3 } 15x^{2 } 5x.
The polynomial (x), if the divisor is 5x^{2}, quotient is 2x + 3, and remainder is 10x + 20 is __________
We know that,
f(x) = q(x) × g(x) + r(x)
Where, f(x) is the dividend, q(x) is the quotient, g(x) is the divisor and r(x) is the remainder.
f(x) = 5x^{2 }× (2x + 3) + 10x + 20
f(x) = 10x^{3 }+ 15x^{2 }+ 10x + 20
What will be the value of a and b if the polynomial f(x) = 30x^{4 } 50x^{3 }+ 109x^{2 } 23x + 25, when divided by 3x^{2 } 5x + 10, gives 10x^{2 }+ 3 as quotient and ax + b as remainder?
We know that,
f(x) = q(x) × g(x) + r(x)
Where, f(x) is the dividend, q(x) is the quotient, g(x) is the divisor and r(x) is the remainder.
∴ 30x^{4 } 50x^{3 }+ 109x^{2 } 23x + 25 = (10x^{2 }+ 3)(3x^{2 } 5x + 10) + ax + b
30x^{4 } 50x^{3 }+ 109x^{2 } 23x + 25 = 30x^{4 } 50x^{3 }+ 109x^{2 } 15x + 30 + ax + b
30x^{4 } 50x^{3 }+ 109x^{2 } 23x + 25  (30x^{4 } 50x^{3 }+ 109x^{2 } 15x + 30) = ax + b
23x + 25 + 15x  30 = ax + b
8x  5 = ax + b
∴ a = 8, b = 5
If α is a zero of the polynomial f(x), then the divisor of f(x) will be _________
If α is a zero of the polynomial f(x).
The divisor will be x  α.
For example, if 5 is a zero of a polynomial f(x), then its divisor will be x  5.
What real number that should be added to the polynomial f(x) = 81x^{2 } 31, so that it is exactly divisible by 9x + 1?
81x^{2 } 31 is exactly divisible by 9x + 1
Hence, on dividing 81x^{2 } 31 by 9x + 1
We get, 9x  1 as quotient and remainder as 30.
So if we add 30 to 81x^{2 } 31, it will be exactly divisible by 9x + 1.
When a polynomial f(x) = acx^{3 }+ bcx + d, is divided by g(x), it leaves quotient as cx, and remainder as d. The value of g(x)will be _____
We know that,
f(x) = q(x) × g(x) + r(x)
Where, f(x) is the dividend, q(x) is the quotient, g(x) is the divisor and r(x) is the remainder.
acx^{3} + bcx + d = cx × g(x) + d
acx^{3} + bcx + d – d = cx × g(x)
g(x) = ax^{2 }+ b
The quotient if the polynomial f(x) = 50x^{2 } 90x  25 leaves a remainder of 5, when divided by 5x  10, will be __________
We know that,
f(x) = q(x) × g(x) + r(x)
Where, f(x) is the dividend, q(x) is the quotient, g(x) is the divisor and r(x) is the remainder.
∴ 50x^{2 } 90x  25 = q(x) × 5x  10  5
50x^{2 } 90x  25 + 5 = q(x) × 5x  10
We get, q(x) = 10x + 2
If two of the zeros of the polynomial f(x) = x^{3 }+ (6  √3)x^{2 }+ (1  √3)x + 30  6√3 are 3 and 2 then, the other zero will be ____________
Since the zeros of the polynomial are 3 and 2.
The divisor of the polynomial will be (x  3) and (x + 2).
Multiplying (x  3) and (x + 2) = x^{2 }+ 2x  3x  6 = x^{2 } x + 6
Dividing, x^{3 }+ (6  √3)x^{2 }+ (1  √3)x + 30  6√3 by x^{2 } x + 6
We get, x  5 + √3 as quotient.
Hence, the third zero will be 5  √3.
If f(x) is divided by g(x), it gives quotient as q(x) and remainder as r(x). Then, f(x) = q(x) × g(x) + r(x) where, f(x) is the dividend, q(x) is the quotient, g(x) is the divisor and r(x) is the remainder.
Consider, f(x) is 27x^{2 } 39x, q(x) as 9x + 2, g(x) as 3x  5 and remainder is 10.
f(x) = q(x) × g(x) + r(x)
RHS
q(x) × g(x) + r(x) = (9x + 2)(3x  5) + 10 = 27x^{2 } 45x + 6x  10 + 10 = 27x^{2 } 39x, which is equal to LHS.
Hence proved.
172 videos94 docs66 tests

Use Code STAYHOME200 and get INR 200 additional OFF

Use Coupon Code 
172 videos94 docs66 tests









