Mathematics · Quantitative Aptitude

Algebraic Expressions and Polynomials

275 Questions

Algebraic expressions and polynomials form the foundation of algebra, involving variables, constants, and mathematical operations. This topic is heavily tested in the quantitative aptitude sections of SSC, banking, and state exams. Use these questions to practice expanding, factoring, and simplifying various polynomial expressions.

Expanding algebraic expressionsFactoring polynomialsAlgebraic identitiesFinding common factorsSimplifying numerical statements

Algebraic Expressions and Polynomials Questions

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

Factorise : $5mn+15mnp$

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

  4. none of these
Reveal answer Fill a bubble to check yourself
A Correct answer
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 $5mn(1+3p)$.

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

If $f(x)$ and $g(x)$ are two polynomials with integral coefficients which vanish at $x = \dfrac {1}{2}$, then what is the factor of HCF of $f(x)$ and $g(x)$?

  1. $x - 1$
  2. $x - 2$
  3. $2x - 1$
  4. $2x + 1$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given, $x = \dfrac {1}{2}$

$ \Rightarrow (2x - 1) = 0$
Therefore, $ (2x - 1)$ is satisfying both $f(x)$ and $g(x)$, so $(2x - 1)$ is the factor of H.C.F. of $f(x)$ and $g(x)$.

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

Consider the following statements :
1. $x - 2$ is a factor of $x^{3} - 3x^{2} + 4x - 4$
2. $x + 1$ is a factor of $2x^{3} + 4x + 6$
3. $x - 1$ is a factor of $x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$
Of these statements

  1. 1 and 2 are correct

  2. 1, 2 and 3 are correct

  3. 2 and 3 are correct

  4. 1 and 3 are correct
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
  1. Remainder $=2^3-3\times 2^2+4\times 2-4$
    $= 8-12+8-4 = 0$
    Hence $x-2$ is a factor.
    2. Remainder$= 2(-1)^3+4(-1)+6$
    $= -2-4+6 = 0$
    Hence $x + 1$ is a factor.
    3. Ramainder $= 1^6-1^5+1^5-1^3+1^2-1+1 = 1$
    Hence $x - 1$ is not a factor.
    $\therefore$ Statements 1 and 2 are correct.
Multiple choice maths algebraic formulae - expansion of squares introduction to factorization introduction to factorisation factorising algebraic expressions

If $4x^{4} -12x^{3}+x^{^{2}}+3ax-b$ is divided by $x^{2}-1$ then a = ____, and b=___

  1. $5, 4$
  2. $4,9$
  3. $4,5$
  4. $1, -1$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

${ x }^{ 2 }-1=0\quad \Rightarrow x=\pm 1$

Given a polynomial $P(x)={ 4x }^{ 4 }-12{ x }^{ 3 }+{ x }^{ 2 }+3ax-b$
Using remainder theorem,as $P(x)$ is completely divisible by ${ x }^{ 2 }-1$
$\therefore P(\pm 1)=0\ \therefore P(1)=0\ \Rightarrow 4-12+1+3a-b=0\ \Rightarrow 3a-b=7\ \therefore P(-1)=0\ \Rightarrow 4+12+1-3a-b=0\ \Rightarrow 3a+b=17$
By soling both we get
$a=4$   ;$b=5$

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

$7+3x$ is a factor of $3x^3+7x$.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

If $7 + 3x$ is a factor of $p\left( x \right) = 3{x^3} + 7x$, then, $p\left( {\frac{{ - 7}}{3}} \right) = 0$.

Compute $p\left( {\frac{{ - 7}}{3}} \right)$ in the given polynomial.

$p\left( {\frac{{ - 7}}{3}} \right) = 3{\left( { - \frac{7}{3}} \right)^3} + 7\left( {\frac{{ - 7}}{3}} \right)$

$ =  - \frac{{343}}{9} - \frac{{49}}{3}$

$ = \frac{{ - 343 - 147}}{9}$

$ =  - \frac{{490}}{9}$

The value of $p\left( {\frac{{ - 7}}{3}} \right)$is not equal to 0.

The given statement is false.

Multiple choice maths be my multiple, i'll be your factor co-prime numbers lcm lowest common multiple (l.c.m.)

Find the common factors of the given terms:

$6 abc, 24ab^2, 12 a^2b$

  1. $6a^2b$
  2. $6ab^2$
  3. $6ab$
  4. $6$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$6 abc$, $24ab^2$, $12 a^2b$
The factors of $6abc=2\times 3\times a\times b\times c$
The factors of $24ab^2=2\times 2\times 2\times 3\times a\times b\times b$
The factors of $12 a^2b=2\times 2\times 3\times a\times a\times b$
The common factors are $2\times 3\times a\times b=6ab$

Multiple choice maths be my multiple, i'll be your factor co-prime numbers lcm lowest common multiple (l.c.m.)

Find the common factors of the given terms:

$16 x^3, 4x^2, 32x$

  1. $x$
  2. $16$
  3. $4x$
  4. $32$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$16 x^3$,  $4x^2$, $32x$
The factors of $16 x^3=2\times 2\times 2\times 2\times x\times x\times x$
The factors of $4x^2=2\times 2 \times x\times x$
The factors of $32x=2\times 2\times 2\times 2\times 2\times x$
The common factors are $2\times 2 \times x=4x$

Multiple choice maths brackets order operations and algebra using brackets in algebraic expressions order of operations

Use the BODMAS rule to reduce the expression: $x-1[(x^2+x-2)(x^2-1^2)\div (x-1)^2]$.

  1. $x^3+2x^2-x-2$
  2. $x^3-2x^2-x-2$
  3. $-x^3+2x^2-x-2$
  4. $x^3+2x^2+x-+$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$x-1[(x^2+x-2)(x^2-1^2)\div (x-1)^2]$
We need to follow BODMAS rule.
=> Brackets (parts of a calculation inside brackets always come first).
=> Orders (numbers involving powers or square roots).
=> Division.
=> Multiplication.
=> Addition.
=> Subtraction.
$=$ $\frac{x-1(x^2+x-2)(x+1)(x-1)}{(x-1(x-1)}$
$=$ $x^3+x^2-2x+x^2+x-2$
$=$ $x^3+2x^2-x-2$

Multiple choice maths brackets order operations and algebra using brackets in algebraic expressions order of operations

Expand the expression using BODMAS rule: $x^2-x[(-x)(-2+x)]\div x+x^3-3x^2$

  1. $x^3-x^2-2x$
  2. $-x^3-x^2-2x$
  3. $x^3-x^2+2x$
  4. $x^3+x^2+2x$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$x^2-x[(-x)(-2+x)]\div x+x^3-3x^2$
We need to follow BODMAS rule.
=> Brackets (parts of a calculation inside brackets always come first).
=> Orders (numbers involving powers or square roots).
=> Division.
=> Multiplication.
=> Addition.
=> Subtraction.
$=$ $-2x^2-x[\dfrac{(-x)(-2+x)}{x}]+x^3$
$=$ $-2x^2-x[2-x]+x^3$
$=$ $x^3-2x^2-2x+x^2$
$=$ $x^3-x^2-2x$

Multiple choice maths brackets order operations and algebra using brackets in algebraic expressions order of operations

Reduce the following expression using BODMAS rule: $2y-1(y-y^2)+5y[(-2y)(y^2-1)]$

  1. $10y^4+11y^2+y$
  2. $-10y^4+11y^2-y$
  3. $-10y^4+11y^2+y$
  4. $-10y^4-11y^2+y$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$2y-1(y-y^2)+5y[(-2y)(y^2-1)]$
We need to follow BODMAS rule.
=> Brackets (parts of a calculation inside brackets always come first).
=> Orders (numbers involving powers or square roots).
=> Division.
=> Multiplication.
=> Addition.
=> Subtraction.
$=$ $2y-y+y^2+5y[-2y^3+2y]$
$=$ $y+y^2-10y^4+10y^2$
$=$ $-10y^4+11y^2+y$

Multiple choice maths brackets order operations and algebra using brackets in algebraic expressions order of operations

$24[x+1]-[x^2-24+x]-[2x^2]\div [x^2]$ using BODMAS rule to reduce the expression.

  1. $x^2+23x+46$
  2. $-x^2+23x+46$
  3. $-x^2-23x+46$
  4. $-x^2+23x-46$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$24[x+1]-[x^2-24+x]-[2x^2]\div [x^2]$
We need to follow BODMAS rule.
=> Brackets (parts of a calculation inside brackets always come first).
=> Orders (numbers involving powers or square roots).
=> Division.
=> Multiplication.
=> Addition.
=> Subtraction.
$=$ $24[x+1]-[x^2-24+x]-[2x^2]\div [x^2]$
$=$ $24x+24-x^2+24-x-2$
$=$ $-x^2+46+23x$
$=$ $-x^2+23x+46$