Factors of Algebraic Expressions and Factorisation

Q. Solve for $x:$

$9^{x+2}=720+9^x$

Q. Factorize$:$
(i) $x^2+5x+6=0$
(ii) $x^2-5x+6=0$
(iii)$x^2+5x-6=0$
(iv) $x^2-5x-6=0$

Q.

Give possible expression for the length and breadth of the rectangle having $35y^2+13y-12$ as its area.

Q. In the wavy curve method , what should we do if we have a not factorizable expression in the denominator ? (or if in the numerator )

Q.

Select the factors of $a^2+7a+10$

1. 1

2. a+1

3. 2

4. a+2

Q. Simplify:

$\left(ii\right)$ $(a+b{)}^{-1}.(a^{-1}+b^{-1})$

Q. factorisation by splitting middle term

Q. Factorize:
$x(x-2)(x-4)+4x-8$

Q. Simplify : ${\left(\frac{x^a}{x^b}\right)}^{a^2+ab+b^2}\times {\left(\frac{x^b}{x^c}\right)}^{b^2+bc+c^2}\times {\left(\frac{x^c}{x^a}\right)}^{c^2+ca+a^2}$

Q.

$x–2$ is a factor of $x^3–2x–4$.

1. True

2. False

Q. Dividend = Quotient $\times$ Divisor + Remainder

1. False
2. True

Q. Dividend = Quotient $\times$ Divisor + Remainder

1. True
2. False

Q. If $a+b+c=0$, then $a^3+b^3+c^3$ is equal to

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

Q. What is the degree of the polynomial
$P\left(x\right)=-5x^9-x^{12}+3+7x^2+10x?$

1. 9
2. 0
3. 10
4. 12

Q. x + y is a factor of .

1. x² - y²
2. x² + y²
3. (x - y)²

Q. Who is correct?
Aakanksha: $q\left(x\right)=\frac{p\left(x\right)-r\left(x\right)}{g\left(x\right)}$
Arjun: $q\left(x\right)=\frac{p\left(x\right)}{g\left(x\right)}-r\left(x\right)$
Aman: $q\left(x\right)=p\left(x\right)-\frac{r\left(x\right)}{g\left(x\right)}$
Anasuya: $q\left(x\right)=p\left(x\right)+\frac{r\left(x\right)}{g\left(x\right)}$

Given,
$p\left(x\right)$ is the dividend,
$g\left(x\right)$ is the divisor,
$q\left(x\right)$ is the quotient, and
$r\left(x\right)$ is the remainder

1. Aakanksha
2. Arjun
3. Aman
4. Anasuya

Q. One of the factors of (81x² – 1) + (1 + 9x)² is

(81x² – 1) + (1 + 9x)² का कोई एक गुणनखण्ड है

1. 81x²
   
2. x²
   
3. x
   
4. 9x – 1
   

Q. The polynomial 5𝑥^2 + 3𝑥 - 5𝑥^2 + 5 is a _____ degree polynomial.

1. first
2. second
3. third
4. fourth

Q. One of the factors of (49x² – 1) + (1 + 7x)² is

(49x² – 1) + (1 + 7x)² का कोई एक गुणनखण्ड है

1. 49x²
2. x²
3. 7x + 1
4. 7x – 1

Q.

Statement 1: A polynomial P(x) of degree greater than or equal to one, when divided by $(x–a)$, leaves a remainder zero.

Statement 2: $(x–a)$ is a factor of the polynomial P(x).

1. Both the statements are true and statement 1 is the correct explanation of statement 2.

2. Both the statements are true but statement 2 is not the correct explanation of statement 1.

3. Statement 1 is true and statement 2 is false

4. Statement 1 is false and statement 2 is true

Q.

If $x^2$ + $y^2$ = 14 and $xy=3$, find the value of 2${(x+y)}^2-5{(x-y)}^2$.

1. 52

2. 14

3. 106

4. 0

Q. Identify the common factor from the expression $9m{n}^2-4m^2{n}^2+7m^{4}n$.

1. $m{n}^2$
2. $m^{2}n$
3. $mn$
4. $m^2{n}^2$

Q.

If $x^2+y^2=14$ and $xy=3$, find the value of $2{(x+y)}^2-5{(x-y)}^2$.

1. 14

2. 52

3. 0

4. 106

Q. Verify whether the indicated numbers are zeros of the polynomials corresponding to them in the following cases:

(iv) $g\left(x\right)=x^3-6x^2+11x-6,$
$x=1,2,3$

Q. The polynomial 5𝑥^2 + 3𝑥 - 5𝑥^2 + 5 is a _____ degree polynomial.

1. first
2. second
3. third
4. fourth

Q. If $a+b+c=8$ $a^2+b^2+c^2=40$. Find the value of $ab+bc+ca$.

1. $10$
2. $12$
3. $14$
4. $16$

Q. $\text{Find the product of}(7x-4y)\text{and}(3x-7y).$

1. $21x^2$ + $28y^2$ + $61xy$
2. $21x^2$ - $28y^2$ - $61xy$
3. $21x^2$ + $28y^2$ - $61xy$
4. $21x^2$ + $28y^2$ - $37xy$

Q.

$x^2$ in $x^2+3x+2$ is its

1. factor

2. multiple

3. term

4. None of the above

Q. This section contains 1 Matrix Match type question, which has 2 Columns (Column I and Column II). Column I has four entries (A), (B), (C) and (D), Column II has four entries (P), (Q), (R) and (S). Match the entries in Column I with the entries in Column II. Each entry in Column I may match with one or more entries in Column II.
इस खण्ड में 1 मैट्रिक्स मिलान प्रकार का प्रश्न है, जिसमें 2 कॉलम (कॉलम I तथा कॉलम II) हैं। कॉलम I में चार प्रविष्टियाँ (A), (B), (C) तथा (D) हैं, कॉलम II में चार प्रविष्टियाँ (P), (Q), (R) तथा (S) हैं। कॉलम I में दी गयी प्रविष्टियों का मिलान कॉलम II में दी गयी प्रविष्टियों के साथ कीजिए। कॉलम I में दी गयी प्रत्येक प्रविष्टि का मिलान कॉलम II में दी गयी एक या अधिक प्रविष्टियों के साथ हो सकता है।

Match the Columns and choose the appropriate answer

कॉलमों का मिलान करें और सही उत्तर चुनें

	Column I कॉलम I		Column II कॉलम II
(A)	If $\sqrt{19-8\sqrt{3}}=a-\sqrt{b},$ then a2 + b2 is यदि $\sqrt{19-8\sqrt{3}}=a-\sqrt{b},$ तब a2 + b2 है	(P)	5
(B)	Remainder when P(x) = x3 – 3x2 + 4x + 21 is divided by (x – 2), is P(x) = x3 – 3x2 + 4x + 21 को (x – 2) द्वारा विभाजित करने पर शेषफल है	(Q)	25
(C)	Ram starts from coordinates A(3, 4) moves to B(7, 4), then goes to C(7, 7). The shortest distance (in units) between A and C is राम निर्देशांक A(3, 4) से B(7, 4) तक चलता है तथा उसके पश्चात् C(7, 7) पर पहुँचता है। तब A व C के मध्य लघुत्तम दूरी (इकाई में) है	(R)	3
(D)	If (k, 2k) is the solution of 4x + 3y = 30, then the value of k is यदि समीकरण 4x + 3y = 30 का हल (k, 2k) है, तब k का मान है	(S)	$\sqrt{625}$

1. A-QS, B-QS, C-P, D-R
2. A-QS, B-QS, C-R, D-P
3. A-PR, B-PR, C-Q, D-S
4. A-R, B-QS, C-Q, D-P

Q. Use property to evaluate: ${13}^3+(-8{)}^3+(-5{)}^3$

1. $1770$
2. $2197$
3. $1331$
4. $1560$