Tag: maths

Questions Related to maths

If the polynomial $f(x)$ is such that $f(-43) = 0$, which of the following is the factor of $f(x)$?

maths algebraic formulae - expansion of squares introduction to factorization introduction to factorisation factorising algebraic expressions

  1. $x - 43$

  2. $x$

  3. $x - 7$

  4. $x + 43$

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Correct Option: D
Explanation:

$f(-43)=0$ 
$\Rightarrow -43$ is root of $f(x)$

$\Rightarrow (x+43).g(x)=f(x)$ for some function $ g(x)$
$\Rightarrow (x+43)$ is the factor of $f(x)$

The denominator of an algebraic fraction should not be

maths algebraic formulae - expansion of squares introduction to factorization introduction to factorisation factorising algebraic expressions

  1. $1$

  2. $0$

  3. $4$

  4. $7$

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Correct Option: B
Explanation:

If denominator of algebraic function is zero then it not defined

Hence denominator of any algebraic fraction should not be $0$

If the sum of two integers is $-2$ and their product is $-24$, the numbers are

maths algebraic formulae - expansion of squares introduction to factorization introduction to factorisation factorising algebraic expressions

  1. $6$ and $4$

  2. $-6$ and $4$

  3. $-6$ and $-4$

  4. $6$ and $-4$

Show answer
Correct Option: B
Explanation:
Let $p$ and $q$ be the required integers

If $p+q$ and $pq$ are known then quadratic equation corresponding to roots as $p$ and $q$ is given by,
$x^2-(p+q)x+pq=0$
$\Rightarrow x^2+2x-24=0$, substitute the given values
$\Rightarrow x^2+6x-4x-24=0$, split the middle term
$\Rightarrow (x^2+6x)+(-4x-24)=0$, group pair of terms
$\Rightarrow x(x+6)-4(x+6)=0$, factor each binomials 
$\Rightarrow (x+6)(x-4)=0$, factor out common factor 
$\Rightarrow x=-6$ or $x=4$, set each factor to $0$

Hence $p$ and $q$ are $-6$ and $4$

The value of  $k$  for which  $x - 1$  is a factor of the polynomial  $4 x ^ { 3 } + 3 x ^ { 2 } - 4 x + k$  is

maths algebraic formulae - expansion of squares introduction to factorization introduction to factorisation factorising algebraic expressions

  1. $3$

  2. $0$

  3. $1$

  4. $-3$

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Correct Option: D
Explanation:

$x-1$ is a factor of $4{x^2} + 3{x^2} - 4x + k$


put $x=1$


$4{x^2} + 3{x^2} - 4x + k=0$

$ \Rightarrow 4{\left( 1 \right)^2} + 3{\left( 1 \right)^2} - 4\left( 1 \right) + k = 0$

$ \Rightarrow 4 + 3 - 4 + k = 0$

$ \Rightarrow k =  - 3$

Hence,
option $(D)$ is correct answer.

Factorise : $6xy^2 + 4x^2y$ 

maths algebraic formulae - expansion of squares introduction to factorization introduction to factorisation factorising algebraic expressions

  1. $2xy(3x+y)$

  2. $xy(3x+2y)$

  3. $2xy(2x+3y)$

  4. none of these

Show answer
Correct Option: C
Explanation:

The common factor between $6xy^2$ and $4x^2y$ is $2xy$ that is the HCF of $6xy^2$ and $4x^2y$ is $2xy$. 

Therefore, we take $2xy$ as a common factor in the expression $6xy^2+4x^2y$ as shown below:
$6xy^2+4x^2y=2xy(2x+3y)$
Hence, the factors of $6xy^2+4x^2y$ are $2xy$ and $(2x+3y)$.

Factorise :
$\displaystyle 121ac-16a^{2}b^{2}$

maths algebraic formulae - expansion of squares introduction to factorization introduction to factorisation factorising algebraic expressions

  1. $a(121c-16ab^{2})$

  2. $a(121c+16ab^{2})$

  3. $a(121c-16ac^{2})$

  4. none of these

Show answer
Correct Option: A
Explanation:

$121ac-16a^2b^2$

$=121\times a\times c-16\times a\times a\times b\times b$
$=a(121c-16ab^2)$

Which of the following is an example of factorisation?

maths algebraic formulae - expansion of squares introduction to factorization introduction to factorisation factorising algebraic expressions

  1. $x^2+2x=x(x+2)$

  2. $x^2+2x=x(x+1)$

  3. $x^2+2x=x(x+3)$

  4. None of the above

Show answer
Correct Option: A
Explanation:

The common factor between $x^2$ and $2x$ is $x$ that is the HCF of $x^2$ and $2x$ is $x$. 


Therefore, we take $x$ as a common factor in the expression $x^2+2x$ as shown below:


$x^2+2x=x(x+2)$


Hence, the factorization of $x^2+2x$ is $x(x+2)$.

Factorise : $5mn+15mnp$

maths algebraic formulae - expansion of squares introduction to factorization introduction to factorisation factorising algebraic expressions

  1. $5mn(1 + 3p)$

  2. $3mn(1 + 5p)$

  3. $5mn(1 - 3p)$

  4. none of these

Show answer
Correct Option: A
Explanation:

The common factor between $5mn$ and $15mnp$ is $5mn$ that is the HCF of $5mn$ and $15mnp$ is $5mn$. 


Therefore, we take $5mn$ as a common factor in the expression $5mn+15mnp$ as shown below:


$5mn+15mnp=5mn(1+3p)$


Hence, the factorization of $5mn+15mnp$ is $5mn(1+3p)$.

Simplify: $\displaystyle \left( -80{ m }^{ 4 }npq \right) \div 10{ m }^{ 3 }{ pqn }^{ 2 }$

maths algebraic formulae - expansion of squares introduction to factorization introduction to factorisation factorising algebraic expressions

  1. $-8mn$

  2. $-8mnpq$

  3. $-8m$

  4. $\dfrac {-8m}{n}$

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Correct Option: D
Explanation:

$\displaystyle \left( -80{ m }^{ 4 }npq \right) \div 10{ m }^{ 3 }{ pqn }^{ 2 }=\frac { -80{ m }^{ 4 }npq }{ 10{ m }^{ 3 }{ pqn }^{ 2 } } $

$\displaystyle =\quad -8\times \frac { { m }^{ 4 } }{ { m }^{ 3 } } \times \frac { n }{ { n }^{ 2 } } \times \frac { p }{ p } \times \frac { q }{ q } $
$\displaystyle =\quad -8{ m }^{ 4-3 }\times { n }^{ 1-2 }$
$\displaystyle =\quad -8m\times { n }^{ -1 }$
$\displaystyle =\quad \frac { -8m }{ n } \left( \because { n }^{ -1 }=\frac { 1 }{ n }  \right) $

The factorisation of $ \left (21a^2+3a \right )$ is

maths algebraic formulae - expansion of squares introduction to factorization introduction to factorisation factorising algebraic expressions

  1. $3a(7a+1)$

  2. $7a(3a+1)$

  3. $3a(7a+3a)$

  4. $3a(a+7)$

Show answer
Correct Option: A
Explanation:

The factorisation of $21a^2+3a$ is $3a(7a+1)$
Taking common terms out, we get $3a(7a+1)$.