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Published on: 26/11/2019
Polynomials
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Questions + Answers key
Take MCQ Mathematics Test
1.
Find the zero of the polynomial in each of the following cases: \(p(x)=2x+5\)
2.
Find p(0), p(1) and p(2) for each of the following polynomials:
p(x) = (x - 1) (x + 1)
3.
Classify the following as linear, quadratic and cubic polynomials:
1 + x
4.
Write the coefficients of x2 in the following:
\(\frac { \pi }{ 2 } { x }^{ 2 }+x\)
5.
Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
\({ x }^{ 10 }+{ y }^{ 3 }+{ t }^{ 50 }\)
6.
If \(x+\frac{1}{x}=7,\) then find the value of \(x^3+\frac{1}{x^3}\)
7.
Factorise: \(a^2+b^2-2(ab-ac+bc)\)
8.
If x and y are two positive real numbers such that \(x^2+4y^2=17\) and xy=2, then find the value of (x+2y).
9.
On dividing \(f(x)={ x }^{ 4 }-{ 2x }^{ 3 }+3{ x }^{ 2 }-ax+b\) and (x+1), we get remainder 5 and 19 respectively. Find the remainder when f(x) is divided by (x-2).
10.
Write the various coefficients in the following polynomials:
1 - 2y + 3y6
11.
Find the value of k if x - 1 is a factor of \(4x^3+3x^2-4x+k\)
12.
If \(p(x)=x^3+3x^2-2x+4\) then find the value of \(p(2)+p(-2)-p(0).\)
13.
Point out which of the following polynomials are monomials, binomials or trinomials?
\({ x }^{ 3 }+8\)
14.
Find the degree of the polynomials given below:
\({ x }^{ 5 }-{ x }^{ 4 }+5\)
15.
Write the coefficient of x3 of the following polynomials:
\({ 2x }^{ 3 }-7x+{ x }^{ 2 }+{ 3x }^{ 4 }\)
16.
Degree of the polynomial \(4x^4+0x^3+0x^5+5x+7\) is:
7
5
4
3
17.
Which of the following is a trinomial in x?
\(x^3+1\)
\(x^3+x^2+x\)
\(x\sqrt{x}-\sqrt {x}+1\)
\(x^3+2x\)
18.
Which of the following is a polynomial in one variable?
\(3-x^2+x\)
\(\sqrt{3x}+4\)
\(x^3+y^3+7\)
\(x+\frac{1}{x}\)
19.
The number 0 is called a
zero polynomial
binomial
trinomial
linear polynomial
20.
The expansion for \((x-y)^2\) is
\((x-y)^2=x^2-2xy+y^2\) is an algebraic identity
\(x^2-2xy+y^2\)
\(x^2+2xy+y^2\)
\(x^2+y^2\)
\(x^2-y^2\)
1.
\(p(x)=2x+5\)
\(p(x)=0\)
\(\\ \Rightarrow \ 2x+5=0\ \Rightarrow \ 2x=-5\ \Rightarrow \ x=-\frac { 5 }{ 2 } \)
\(\therefore \ -\frac { 5 }{ 2 } \) is a zero of the polynomial p(x)
2.
\(\therefore \ p(0)=(0-1)(0+1)=(-1)(1)=-1\)
\(\\ p(1)=(1-1)(1+1)=(0)(2)=0\)
and \(p(2)=(2-1)(2+1)=(1)(3)=3\)
3.
linear
4.
Coefficient of \({ x }^{ 2 }=\frac { \pi }{ 2 } \)
5.
This expression is not a polynomial in one variable because in the expression, three variables (x, y and t) occur.
6.
We know that
\({ \left( x+\frac { 1 }{ x } \right) }^{ 3 }={ x }^{ 3 }+\frac { 1 }{ { x }^{ 3 } } +3(x)\left( \frac { 1 }{ x } \right) \left( x+\frac { 1 }{ x } \right) \)
\(\Rightarrow { \left( x+\frac { 1 }{ x } \right) }^{ 3 }={ x }^{ 3 }+\frac { 1 }{ { x }^{ 3 } } +3\left( x+\frac { 1 }{ x } \right) .\)
\(\Rightarrow { (7) }^{ 3 }={ x }^{ 3 }+\frac { 1 }{ { x }^{ 3 } } +3(7)\)
\(\Rightarrow { x }^{ 3 }+\frac { 1 }{ { x }^{ 3 } } =322\)
7.
\(a^2+b^2-2(ab-ac+bc)\)
\(=a^2+b^2-2ab+2ac-2bc\)
\(=(a-b)^2+2c(a-b)\)
\((a-b)(a-b+2c).\)
8.
We know that
\((x+2y)^2=(x)^2+(2y)^2+2(x)(2y)\)
| Using Identity I
=\(x^2+4y^2+4xy=17+4(2)\)
\(=17+8=25\)
\(\Rightarrow (x+2y)=5\)
9.
\(f(x)={ x }^{ 4 }-{ 2x }^{ 3 }+3{ x }^{ 2 }-ax+b\)
By remainder theorem,
\(f(1)=5 |x-1=0\quad \Rightarrow x=1\)
\(\Rightarrow { (1) }^{ 4 }-2{ (1) }^{ 3 }+3{ (1) }^{ 2 }-a(1)+b=5\)
\(\Rightarrow 1-2+3-a-b=5\)
\(\Rightarrow -a+b+2=5\)
\(\Rightarrow a-b=-3\ \ .........(1)\)
and\(f(-1)=19\quad |\quad x+1=0\quad \Rightarrow x=-1\)
\(\Rightarrow { (-1) }^{ 4 }-2{ (-1) }^{ 3 }+3{ (-1) }^{ 2 }-a(-1)+b=19\)
\(\Rightarrow1+2+3+a+b=19\)
\(\Rightarrow a+b=13\ \ .......(2)\)
Solving (1) and (2), we get a=5, b=8
\(\therefore f(x)\ ={ x }^{ 4 }-{ 2x }^{ 3 }+3{ x }^{ 2 }-5x+8\)
\(\therefore \) Remainder when f(x) is divided by x-2 = f(2)
By remainder theorem: x-2=0\(\Rightarrow \) x=2
\( =16-16+12-10+8\)
= 10
10.
1, -2, 3
11.
As x – 1 is a factor of p(x) = 4x3 + 3x2 – 4x + k, p(1) = 0
Now, p(1) = 4(1)3 + 3(1)2 – 4(1) + k
So, 4 + 3 – 4 + k = 0
i.e., k = –3
12.
28
13.
binomial
14.
5
15.
2
16.
Highest power of x=4
17.
\(x^3+x^2+x\) has three terms
18.
Fractional power of x in (b),
Two variables in (c),
Negative power of x in (d)
19.
Definition of zero polynomial
20.
(a)
\(x^2-2xy+y^2\)
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