Long Division of Polynomials

Q. If the polynomial $f\left(x\right)=a{x}^3+bx-c$ is divisible by the polynomial $g\left(x\right)=x^2+bx+c$, then $ab=?$

1. $-\frac{1}{c}$
2. -1
3. 1
4. $\frac{1}{c}$

Q. Using synthetic division, solve $(x^4+2x^3+3x^2+4x+5)÷(x+2)$.
Write the quotient and the remainder.

Q.

If $(x^2+px+1)$ is a factor of $(a{x}^3+bx+c)$, then

1. $a^2+c^2=-ab$

2. $a^2-c^2=-ab$

3. $a^2-c^2=ab$

4. None of these

Q.

If $x^3+2x+1$is divided by $x-2$, then the remainder is:

1. 13

2. 12

3. 10

4. None of these

Q. Dividend = Quotient x Remainder + Divisor

1. False
2. True

Q. $–4x^3–6x^2+8x\text{by}2x.$

Q. The remainder obtained when ($x^4+4x^3+3$) is divided by ($x+1$) =

1. \N
2. 1
3. 2

Q. The quotient when $4x^3+x^2+5x+8$ is divided by $(x+1)$ is .

1. $(4x^2+3x–8)$
2. $(x^2–3x+8)$
3. $(4x^2–3x+8)$

Q. Select the terms that are used in the "long division process".

1. Divisor
2. Flower bracket
3. Quotient
4. synthetic
5. Dividend
6. Divider
7. Remainder
8. Open bracket

Q.

Find the quotient and remainder for each of the following.

$(-132)÷5$

Q.

When we divide $3t^4+5t^3-7t^2+2t+2$ by $t^2+3t+1$, we get $3t^2-4t+2$ as quotient.

1. False

2. True

Q.

The factors of the polynomial equation P(x) = $4x^3+x^2+5x+8$ are

1. $(x+1)$, $(4x^2–3x+8)$
2. $(x+2)$, $(x^2–3x+8)$
3. $(x+3)$, $(x^2+8x–3)$
4. $(x+4)$, $(4x^2+3x–8)$

Q.

$\text{Given the area of rectangle is}A=25a^2-35a+69.\text{The length is given}\text{as}(5a-3).\text{Find the width of the}\text{rectangle.}$

1. $5a-3$

2. $4a-5$

3. $a-4$

4. $5a-4$

Q. When $x^4+x^3-2x^2+x+1$ is divided by $x-1$, the remainder is 2 and the quotient is q(x). Find q(x).

1. $x^3+2x^2+x+1$
2. $x^3+x^2+x$
3. $x^3+2x^2+1$
4. $x^3+x^2+1$

Q.

$p\left(x\right)=x^3+8x^2-7x+12$ and $g\left(x\right)=x-1$. If $p\left(x\right)$ is divided by $g\left(x\right)$, it gives $q\left(x\right)$ and $r\left(x\right)$ as quotient and remainder respectively. If $a$ is the degree of $q\left(x\right)$ and $b$ is the degree of $r\left(x\right)$, $(a-b)=?$.

1. 1

2. 2
3. 3
4. 4

Q.

Divide $f\left(x\right)$ by $g\left(x\right)$ and find the quotient.
$f\left(x\right)=x^3-125,g\left(x\right)=x^2+5x+25$

1. x + 5

2. x - 5

3. $x^2+5x$

4. x + 2

Q. $\text{If}(x-3)\text{and}(x-2)\text{are two factors of}\text{the polynomial}{x}^3-6x^2+11x-6,\text{then find the third factor.}$

1. $x-6$
2. $x-5$
3. $x-1$
4. $x-4$

Q.

$\text{Given the area of rectangle is}A=25a^2-35a+69.\text{The length is given}\text{as}(5a-3).\text{Find the width of the}\text{rectangle.}$

1. $5a-3$

2. $5a-4$

3. $4a-5$

4. $a-4$

Q. Dividend = Quotient x Remainder + Divisor

1. True
2. False

Q.

How do you use synthetic division to find $f\left(-3\right)$ for $f\left(x\right)=x^4-4x^3+2x^2-4x+6$?

Q. Using factor theorem, factorize the polynomial $x^3-6x^2+11x-6$.