This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Division of Polynomial”.

1. 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.

a) True

b) False

View Answer

Explanation: 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.

2. If α is a zero of the polynomial f(x), then the divisor of f(x) will be _________

a) x<α

b) x-α

c) x>α

d) x+α

View Answer

Explanation: 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.

3. 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 ____________

a) -√3

b) 5

c) 5-√3

d) 5+√3

View Answer

Explanation: 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.

4. 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?

a) a=8, b=5

b) a=-8, b=5

c) a=8, b=-5

d) a=-8, b=-5

View Answer

Explanation: 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

5. The quotient if the polynomial f(x)=50x^{2}-90x-25 leaves a remainder of -5, when divided by 5x-10, will be __________

a) 10x+2

b) 10x-2

c) -10x+2

d) -10x-2

View Answer

Explanation: 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

\(\frac {50x^2-90x-20}{5x-10}\)=q(x)

We get, q(x)=10x+2

6. The polynomial (x), if the divisor is 5x^{2}, quotient is 2x+3, and remainder is 10x+20 is __________

a) 10x^{3}-15x^{2}-10x-20

b) -10x^{3}-15x^{2}+10x+20

c) 10x^{3}+15x^{2}+10x+20

d) -10x^{3}+15x^{2}+10x+20

View Answer

Explanation: 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

7. 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 _____

a) -ax^{2}+b

b) ax^{2}-b

c) ax^{2}+b

d) x^{2}+b

View Answer

Explanation: 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)

\(\frac {acx^3+bcx}{cx}\)=g(x)

g(x)=ax

^{2}+b

8. 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 ____________

a) -12

b) -11

c) 11

d) 12

View Answer

Explanation: 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.

9. What real number that should be added to the polynomial f(x)=81x^{2}-31, so that it is exactly divisible by 9x+1?

a) 40

b) 10

c) 30

d) 20

View Answer

Explanation: 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.

10. If the polynomial f(x)=x^{2}+kx-15, is exactly divisible by x-5, then the value of k is _______

a) 3

b) 2

c) -3

d) -2

View Answer

Explanation: 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

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