Polynomials

Q.

If $x+\frac{1}{x}=7$ then find the value of $x^3+\frac{1}{x^3}$

Q. If a+b+c is equal to 0 , then find the value of a cube + b cube + c cube

Q. For what value of $m$ is $x^3–2m{x}^2+16$ is divisible by x + 2?

1. 
   $0$
2. 
   $1$
3. 
   $2$
4. 
   $3$

Q.

If 125^x=25/5 ^{​​​​​x}find x.

Q.

Which of the following expressions are polynomials? In case of a polynomial, write its degree.

(i) $x^5-2x^3+x+\sqrt{3}$

(ii) $y^3+\sqrt{3}y$

(iii) $t^2-\frac{2}{5}t+\sqrt{5}$

(iv) $x^{1000}-1$

(v) $\frac{1}{\sqrt{2}}{x}^2-\sqrt{2}x+2$

(vi) $x^{-2}+2x^{-1}+3$

(vii) $1$

(viii) $\frac{-3}{5}$

(ix) $\frac{x^2}{2}-\frac{2}{x^2}$

(x) $\sqrt[2]{22}{x}^2-8$

(xi) $\frac{1}{2x^2}$

(xii) $\frac{1}{\sqrt{5}}{x}^{\frac{1}{2}}+1$

(xiii) $\frac{3}{5}{x}^2-\frac{7}{3}x+9$

(xiv) $x^4-x^{\frac{3}{2}}+x-3$

(xv) $2x^3+3x^2+\sqrt{x}-1$

Q.

If 9^n×3^2×3^n-(27)^n÷3^3m×2^3=1/27, prove that m=1+n.

Q. The expression (a − b)³ + (b − c)³ + (c −a)³ can be factorized as

(a) (a − b) (b − c) (c −a)

(b) 3(a − b) (b − c) (c −a)

(c) −3(a − b) (b −c) (c − a)

(d) (a + b + c) (a² + b² + c² − ab − bc − ca)

Q. x³ + 6x² + 12x + 16

Q.

Why x raise to the power 0 is equal to 1?

Q.

Factorise

a⁶+b⁶

Q.

Factories $y^2-5y+6$ by using splitting middle term.

Q.

Identify polynomials in the following :

(i) f(x) = $4x^3-x^2$ - 3x + 7

(ii) g(x) = $2x^3-3x^2+\sqrt{x}-1$

(iii) p(x) = $\frac{2}{7}{x}^2-\frac{7}{4}x+9$

(iv) q(x) = $2x^2-3x+\frac{4}{x}+2$

(v) h(x) = $x^4-x^{\frac{3}{2}}+x-1$

(vi) f(x) = 2 +$\frac{3}{x}+4x$

Q.

If $p\left(x\right)=x+4$ then $p\left(x\right)+p(-x)=?$
(a) 0
(b) 4
(c) 2x
(d) 8

Q.

If $p\left(x\right)=5-4x+2x^2$, find

(i) p(0)

(ii) p(3),

(iii) p(-2)

Q.

Define value of polynomial at a point.

Q.

If $p\left(y\right)=4+3y-y^2+5y^3$, find
(i) p(0),
(ii) p(2),
(iii) p(-1)

Q.

What is the degree of a constant term?

Q. Find the values of a and b, if x² − 4 is a factor of ax⁴ + 2x³ − 3x² + bx − 4

Q. [5(8^1/3+27^1/3)^3]^1/4

Q.

Find x if 27^x =9/3^x

Q.

Solve: $x^3-3x^2-9x-5$

Q.

Factorise:

25a^2-35a+12

Q. x³ − 23x² + 142x − 120

Q. If x³ + ax² − bx+ 10 is divisible by x² − 3x + 2, find the values of a and b.

Q. Factorize
(p-q)³ + (q-r)³ + (r-p)³

Q. Is 5 a polynomial or not? State reason

Q.

Factorise by splitting the middle term: 24x² - 65x + 21

1. (3x- 7) ( 8x + 3)

2. (3x - 7) ( 8x - 3)

3. (3x + 7) ( 8x + 3)

4. (3x +7) ( 8x - 3)

Q.

Find $p\left(0\right),p\left(1\right)$ and $p\left(2\right)$ for the following polynomial: $p\left(x\right)=(x-1)(x+1)$

Q. The expression x⁴ + 4 can be factorized as

(a) (x² + 2x + 2) (x² − 2x + 2)

(b) (x² + 2x + 2) (x² + 2x + 2)

(c) (x² − 2x − 2) (x² − 2x + 2)

(d) (x² + 2) (x² − 2)

Q.

What is the coefficient of $x^2$in $3x^3+2x^2-x+1$?