Division of a Polynomial by a Monomial

Q. Question 14
By Remainder theorem, find the remainder when p(x) is divided by g(x).

(i) $p\left(x\right)=x^3–2x^2–4x–1,g\left(x\right)=x+1$
(ii) $p\left(x\right)=x^3–3x^2+4x+50,g\left(x\right)=x–3$
(iii) $p\left(x\right)=4x^3–12x^2+14x–3,g\left(x\right)=2x–1$
(iv) $p\left(x\right)=x^3–6x^2+2x-4,g\left(x\right)=1-\frac{3}{2}x$

Q. Question 7
If a, b and c are all non-zero and a + b + c = 0, then prove that $\frac{a^2}{bc}+\frac{b^2}{ac}+\frac{c^2}{ab}=3$.

Q. Question 21
Find the value of m, so the 2x- 1 be a factor of $8x^4+4x^3–16x^2+10x+m$.

Q. Factorise:
(i) $12x^2-7x+1$
(ii) $2x^2+7x+3$

Q.

Factorise $27-x^3{y}^3+6-2xy$

1. $(2-xy)(9-3xy+x^2{y}^2)$

2. $(2-xy)(11-3xy+x^2{y}^2)$

3. $(2-xy)(11-4xy)$

4. $(2+xy)(11-3xy-x^2{y}^2)$

Q.

Factorise each of the following: 8$a^3$ – $b^3$– 12$a^2$b + 6a$b^2$

Q. Add the following polynomials.
$\left(i\right)x^2-9x+\sqrt{3}:-19x+\sqrt{3}+7x^2$
$\left(ii\right)2a{b}^2+3a^{2}b-4ab;3ab-8a{b}^2+2a^{2}b$

Q. Find the remainder using remainder theorem, when:

Q. If $A=2y^2+3x-x^2,B=3x^2-y^2$ and $C=5x^2-3xy$ then find
$B+C$

Q. Factorise : $16-x^2-2xy-y^2$ The factors are

1. $(4+x-y)(4+x+y)$
2. $(4-x-y)(4-x+y)$
3. $(2+x+y)(2-x-y)$
4. $(4+x+y)(4-x-y)$

Q. If the polynomial $x^3-a{x}^2+2x-a$ is divided $x-a,$ then remainder is:

1. $a^3$
2. $a^2$
3. $a$
4. $-a$

Q.

$\text{Divide}{m}^2+7m-60\text{by}(m-5).$

1. 4(m - 6)

2. 2(m + 6)

3. (m + 12)

4. (m - 12)

Q. Factorise:
(i) $12x^2-7x+1$
(ii) $2x^2+7x+3$

Q. $\frac{27x^{3}y}{9x^2}=$ ______.

1. $9x^{2}y$
2. $9x{y}^2$
3. $9xy$
4. $3xy$

Q. Find the remainder when $4x^3-5x^2+6x-2$ is divided by $x-1$.

Q. Factorise:
(i) $12x^2-7x+1$
(ii) $2x^2+7x+3$

Q. Divide $2x^3+3x^2-11x-6$ by $x^2+x-6$.

Q. Find the value of $m,$ if $2^{m+6}={16}^m$.

1. 2

Q. In each of the following, using the remainder theorem, find the remainder when $f\left(x\right)$ is divided by $g\left(x\right)$ and verify the result by actual division:

1. $f\left(x\right)=x^3+4x^2-3x+10,g\left(x\right)=x+4$

Q. In each of the following, use factor theorem to find whether polynomial $g\left(x\right)$ is a factor of polynomial $f\left(x\right)$ or not:

$\left(2\right)$ $\begin{array}{c}f\left(x\right)=3x^4+17x^3+9x^2-\\ 7x-10;g\left(x\right)=x+5\end{array}$

Q.

Factorise $a^2+7a+10$ + pa + 6p

1. a+1

2. 2

3. a+2

4. 1

Q.

The factorization of $x^3$+1 is

1. (x+1)(x²+x+1)

2. (x+1)(x²+x-1)

3. (x+1)(x²-x+1)

4. (x-1)(x²-x+1)

Q.

Factorise $(n^2+22n+96)$

1. (n + 16)(n + 6)

2. (n – 16)(n + 6)

3. (n – 16)(n – 6)

4. (n + 16)(n – 6)

Q. Use the Factor Theorem to determine whether g(x) is a factor of p(x) if $p\left(x\right)=2x^3+x^2-2x-1$ and $g\left(x\right)=x+1$

Q. $y^3-64;y-4$

1. $(y-4)(y^2+4y+16)+0$
2. $(y+2)(y^2+y+16)+0$
3. $(y-3)(y^2+3y+2)+0$
4. $(y+74)(y^2-4y+16)+0$

Q.

Factorise $2x^2+13x+15$

1. (x+ 5) (2x - 3)

2. (x+ 5) (2x + 3)

3. ( x - 5) ( 2x - 3)

4. (x- 5) (2x + 3)

Q.

$Solve:4yz(z^2+6z-16)÷2y(z+8)$

1. $z(z-4)$

2. $z-2$

3. $z(z-2)$

4. $2z(z-2)$

Q. If the degree of $12x^3{y}^8{z}^n$ is $14$, then $n=$

Q. The obtained polynomial when $3x^2+x$ is divided by $x$ is:

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

Q.

Factorising and dividing $a^2+7a+10$ by $(a+5)$ gives:

1. a + 1

2. 1

3. 2

4. a + 2