Factor of Polynomials

Q. Question 9
Prove that $(a+b+c{)}^3–a^3–b^3–c^3=3(a+b\left)\right(b+c\left)\right(c+a)$.

Q. X+y=10 and xy = 21, find 2( x^{​​​​​​2 _{+ Y} 2)}

Q.

If $a+b+c=7$ and $ab+bc+ca=20$ find the value of $a^2+b^2+c^2$

Q.

(a²-b²)(c²-d²)-4abcd

Q.

Prove that

2a^2+2b^2+2c^2-2ab-2bc-2ca={(a-b)^2+(b-c)^2+(c-a)^2}

Q. f($x$) = $x^2+kx+6$.
If ($x-2$) is a factor of f($x$), then the value of k = .

1. 5
2. -5
3. 4
4. -4

Q.

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(i) $x^2-2x-8$

(ii) $4s^2-4s+1$

(iii) $6x^2-3-7x$
(iv) $4u^2+8u$

(v) $t^2-15$

(vi) $3x^2-x-4$

Q.

find the value of x2+y2+z2 if x+y+Z=9 and XY+YZ+ZX=26

Q.

_{Simplify (√5-2)^2}

Q.

find the product:(x+y+2z) (x²+y²+4z²-xy-2yz-2zx

Q.

‘p’ and ‘q’ are the zeroes of the polynomial
$a{x}^2+bx+c$. The values of $a$ and $b$ are

1. -1, -(p + q)
2. 1, p + q
3. 1, -(p+q)
4. -1, -(p+q)

Q. $\text{Check whether}{x}^4+3x^3+3x^2+x+1\text{has}\text{(x + 1) as a factor.}$

Q. Factorize a³-64b³-27c³-36abc

Q. Question 1 (iv)
Find the zeroes of the following polynomials by factorization method and verify the relations between the zeroes and the coefficient of the polynomials
(iv) $t^3–2t^2–15t$

Q.

Can you tell me the derivation of identity (a + b)3 = a3 + b3 + 3ab (a + b)

Q. Expand-: (-2a+3b+4c)

Q.

If $ab+bc+ca=36$ and $a^2+b^2+c^2=85$ then find $a+b+c$.

Q. If (mn – 1)² = (mn)² + 1 + 2y, then y = ____ .

1. 2n
2. mn
3. -mn
4. m

Q. Question 3
if both x – 2 and $x-\frac{1}{2}$ are factors of $p{x}^2+5x+r$, then show that p = r.

Q. If $a+b+c=8$ and $ab+bc+ca=20$, then find the value of $a^3+b^3+c^3–3abc$.

1. 32

Q.

2x(a-b) + 3y(5a -5b) + 4z(2b -2a) we have to take out the common factors

Q.

If a not equal to b and x²+ax+b and x²+bx+a have a common factor, show that a+b+1=0

Q. Question 30
if $a+b+c=9$ and $ab+bc+ca=26$, find $a^2+b^2+c^2$.

Q. Question 3
if both x – 2 and $x-\frac{1}{2}$ are factors of $p{x}^2+5x+r$, then show that p = r.

Q.

Find the value of using suitable identity

1. 9999

2. 9998

3. 8999

4. 9989

Q.

‘p’ and ‘q’ are the zeroes of the polynomial $a{x}^2+bx+c=0$.

(x-p) and (x-q) are the only factors of the expression $a{x}^2+bx+c$,

The values of $a$ and $b$ are:

1. 1, p + q

2. 1, -(p+q)

3. -1, -(p+q)

4. -1, -(p + q)

Q.

Find the value of 'a' if (x-2) is factor of 2x³ − 6x² + 5x + a.

Q. For what value of $k$, is $3$ a zero of the polynomials $2x^2+x+k$

Q.

If a² + b² + c² = 250 and ab + bc + ca = 3 find a + b + c.

1. 

2. 

3. 

4. 

Q.

Factorize :

(a + b + c)² - ( a- b - c)² + 4b² - 4c²