Question 11 Mark
Write the degree of the polynomial for each of the following: $ax ^7+ bx ^9$ ( $a , b$ are constants)
Answer$a x^7+b x^9$
Here, the highest power ofx is 9 .
$\therefore$ Degree of the polynomial $=9$
View full question & answer→Question 21 Mark
Write the degree of the polynomial for each of the following : $7=7 x ^0$
Answer$7=7 x^{\circ}$
$\therefore$ Degree of the polynomial $=0$
View full question & answer→Question 31 Mark
Write the degree of the polynomial for each of the following: $5+3 x^4$
Answer$5+3 x^4$
Here, the highest power of $x$ is 4 .
$\therefore$ Degree of the polynomial $=4$
View full question & answer→Question 41 Mark
Factorize : $6 x^2-5 x-6$
Answer$=6 x^2-9 x+4 x-6$
$=3 x(2 x-3)+2(2 x-3)$
$=(2 x-3)(3 x+2)$
View full question & answer→Question 51 Mark
Factorize: $63 x^2+5 x-2$
Answer$=63 x^2+14 x-9 x-2$
$=7 x(9 x+2)-1(9 x+2)$
$=(9 x+2)(7 x-1)$
View full question & answer→Question 61 Mark
Factorize: $3 x^2+7 x+2$
Answer$=3 x^2+6 x+x+2$
$=3 x(x+2)+1(x+2)$
$=(x+2)(3 x+1)$
View full question & answer→Question 71 Mark
Factorize: $4 x^2-25$
Answer$4 x^2-25$
$=(2 x)^2-(5)^2$
$=(2 x+5)(2 x-5)$
View full question & answer→Question 81 Mark
If $(x-1)$ is the factor of the polynomial $\left(x^3-2 x^2+m x-4\right)$ then find the value of $m$.
Answer( $x -1$ ) is factor of $p(x), \therefore p(1)=0$
$p(x)=x^3-2 x^2+m x-4$
$p (1)=1^3-2 \times 1^2+ m \times 1-4=0$
$\therefore 1-2 \times 1+m-4=0$
$\therefore 1-2+ m -4=0 \therefore m-5=0 \therefore m=5$
View full question & answer→Question 91 Mark
Check whether, $x-2$ is a factor of the polynomial $x^3-x^2-4$ by using factor theorem.
Answer$p(x)=x^3-x^2-4 \text { Divisor }=x-2$
$therefore p(2)=2^3-2^2-4=8-4-4=0$
$\therefore$ By factor theorem $(x-2)$ is a factor of the polynomial $\left(x^3-x^2-4\right)$.
View full question & answer→Question 101 Mark
If $p(x)=x^3+4 x-5$ is divided by $x+2$ then find the remainder and hence determine whether $(x+2)$ is a factor of $p(x)$ or not.
View full question & answer→Question 111 Mark
If $p(x)=\left(x^3+4 x-5\right)$ is divided by $(x-1)$ then find the remainder and hence determine whether $(x-1)$ is a factor of $p(x)$ or not?
View full question & answer→Question 121 Mark
By using remainder theorem divide the polynomial $x ^3-2 x ^2-4 x -1$ by $x -1$ and find the remainder.
View full question & answer→Question 131 Mark
If the value of the polynomial $m^2-a m+7$ for $m=-1$ is 10 , then find the value of $a$.
View full question & answer→Question 141 Mark
If $p(x)=2 x^2-x^3+x+2$ then find $p(0)$.
View full question & answer→Question 151 Mark
Find the value of $p(y)=2 y^3-2 y+\sqrt{7}$ for $y=-2$
View full question & answer→Question 161 Mark
Find the value of the polynomial p(x) = $2x^2 -3x + 5$ for $x = 2.$
View full question & answer→Question 171 Mark
Multiply : $-2 a \times 5 a^2$
Answer$-2 a \times 5 a^2=-10 a^3$
View full question & answer→Question 181 Mark
$(2,-1,0,5,6)$ is the coefficient form of the polynomial. Represent it in index form.
AnswerCoefficient form of the polynomial is $(2,-1,0,5,6)$
$\therefore$ index form of the polynomial is $2 x^4-x^3+0 x^2+5 x+6$ i.e. $2 x^4-x^3+5 x+6$
View full question & answer→Question 191 Mark
Write the polynomial $x^3+3 x-5$ in coefficient form.
Answer$\quad x^3+3 x-5=x^3+0 x^2+3 x-5$
$\therefore$ given polynomial in coefficient form is $(1,0,3,-5)$
View full question & answer→Question 201 Mark
Write the degree of the given polynomials : $m^3n^7 – 3m^5n + mn$
Answer$m^3n^7 – 3m^5n + mn$
Here, the sum of the powers of m and n in the term $m^3n^7$ is $3 + 7 = 10,$
which is the highest sum of powers in the given polynomial.
∴ Degree of the polynomial = 10
View full question & answer→Question 211 Mark
Write the degree of the given polynomials : xyz +xy-z
Answerxyz + xy – z
Here, the sum of the powers of x, y and z in the term xyz is 1 + 1 + 1= 3,
which is the highest sum of powers in the given polynomial.
∴Degree of the polynomial = 3
View full question & answer→Question 221 Mark
Write the degree of the given polynomials : $7y – y^3 + y^5$
Answer$7y – y^3 + y^5$
Here, the highest power of y is 5.
∴Degree of the polynomial = 5
View full question & answer→Question 231 Mark
Write the degree of the given polynomials : 2p – √7
Answer2p – √7
Here, the highest power of p is 1.
∴ Degree of the polynomial = 1
View full question & answer→Question 241 Mark
Write the degree of the given polynomials : $√2m^{10} – 7$
Answer$√2m^{10} – 7$
Here, the highest power of m is $10$.
∴Degree of the polynomial = $10$
View full question & answer→Question 251 Mark
Write the degree of the given polynomials : $x^2$
Answer$x^2$
∴Degree of the polynomial = $2$
View full question & answer→Question 261 Mark
Write the degree of the given polynomials : x°
Answerx°
∴Degree of the polynomial = 0
View full question & answer→Question 271 Mark
Write the degree of the given polynomials : √5
Answer√5 = √5 x°
∴ Degree of the polynomial = 0
View full question & answer→Question 281 Mark
Write the polynomial in $x$ using the given information : Trinomial with degree 8
Answer$3x^8 + 2x^6 + x^5$
View full question & answer→Question 291 Mark
Write the polynomial in $x$ using the given information : Binomial with degree $35$
View full question & answer→Question 301 Mark
Write the polynomial in $x$ using the given information : Monomial with degree $7$
View full question & answer→Question 311 Mark
Write the coefficient of $m ^3$ in each of the given polynomial : $\sqrt[-2]{3} m^3+5 m^2-7 m-1$
View full question & answer→Question 321 Mark
Write the coefficient of $m ^3$ in each of the given polynomial : $\sqrt[-3]{2}+m-\sqrt{3} m^3$
View full question & answer→Question 331 Mark
Write the coefficient of $m ^3$ in each of the given polynomial : $m ^3$
View full question & answer→Question 341 Mark
State whether the given algebraic expressions are polynomials? Justify : 10
AnswerYes, because 10 is a constant polynomial.
View full question & answer→Question 351 Mark
State whether the given algebraic expressions are polynomials? Justify : $2 m^{-2}+7 m-5$
AnswerNo, because the power of $m$ in the term $2 m^{-2}$ is -2 (negative number).
View full question & answer→Question 361 Mark
State whether the given algebraic expressions are polynomials? Justify : $x^2+7 x+9$
AnswerYes. All the coefficients are real numbers. Also, the power of each term is a whole number.
View full question & answer→Question 371 Mark
State whether the given algebraic expressions are polynomials? Justify : $2-5 \sqrt{ } x$
AnswerNo, because the power of x in the term 5√x is , i. e. 0.5 (decimal number).
View full question & answer→Question 381 Mark
State whether the given algebraic expressions are polynomials? Justify : $y+\frac{1}{y}$
AnswerNo, because power of v in the term 5√x is -1 (negative number).
View full question & answer→Question 391 Mark
Give example of a binomial in two variables having degree $5.$
View full question & answer→Question 401 Mark
Write an example of a monomial, a binomial and a trinomial having variable x and degree 5.
AnswerMonomial: $x^5$
Binomial: $x^5+x$
Trinomial: $2 x^5-x^2+5$
View full question & answer→Question 411 Mark
Which polynomial is to be added to 4m + 2n + 3 to get the polynomial 6m + 3n + 10?
AnswerLet the required polynomial be A.
∴ (4m + 2n + 3) + A = 6m + 3n + 10
∴ A = 6m + 3n + 10 – (4m + 2n + 3)
= 6m + 3n + 10 – 4m – 2n – 3
= 6m – 4m + 3n – 2n + 10 – 3
= 2m + n + 7
∴ 2m + n + 7 must be added to 4m + 2n + 3 to get 6m + 3n + 10.
View full question & answer→Question 421 Mark
Which polynomial is to be subtracted from $x^2+13 x+7$ to get the polynomial $3 x^2+5 x-4$ ?
AnswerLet the required polynomial be A .
$\therefore\left(x^2+13 x+7\right)-A=3 x^2+5 x-4$
$\therefore A=\left(x^2+13 x+7\right)-\left(3 x^2+5 x-4\right)$
$=x^2+13 x+7-3 x^2-5 x+4$
$=x^2-3 x^2+13 x-5 x+7+4$
$=-2 x^2+8 x+11$
$\therefore-2 x^2+8 x+11$ must be subtracted from $x^2+13 x+7$ to get $3 x^2+5 x-4$.
View full question & answer→Question 431 Mark
Simplify.
$\left(8 m^2+3 m-6\right)-(9 m-7)+\left(3 m^2-2 m+4\right)$
Answer$\left(8 m^2+3 m-6\right)-(9 m-7)+\left(3 m^2-2 m+4\right)$
$=8 m^2+3 m-6-9 m+7+3 m^2-2 m+4$
$=8 m^2+3 m^2+3 m-9 m-2 m-6+7+4$
$=11 m^2-8 m+5$
View full question & answer→Question 441 Mark
Polynomials $b x^2+x+5$ and $b x^3-2 x+5$ are divided by polynomial $x-3$ and the remainders are m and n respectively. If $m-n=0$, then find the value of $b$.
AnswerWhen polynomial $b x^2+x+5$ is divided by $(x-3)$, the remainder is $m$.
$\therefore$ By remainder theorem,
$\text { Remainder }=p(3)=m$
$p(x)=b x^2+x+5$
$\therefore p(3)=b(3)^2+3+5$
$\therefore m=b(9)+8$
$m=9 b+8 \ldots \text { (i) }$
When polynomial $b x^3-2 x+5$ is divided by $x-3$ the remainder is $n$
$\therefore \text { remainder }=p(3)=n$
$p(x)=bx x^3-2 x+5$
$\therefore P(3)=b(3)^3-2(3)+5$
$\therefore n=b(27)-6+5$
$\therefore n=27 b-1 \ldots \text { (ii) }$
Now, $m - n =0 \ldots$ [Given]
$\therefore m=n$
$\therefore 9 b+8=27 b-1 \ldots[\text { From (i) and (ii) }]$
$\therefore 8+1=27 b-9 b$
$\therefore 9=18 b$
$\therefore b=\frac{1}{2}$
View full question & answer→Question 451 Mark
At the end of the year $2016$, the population of villages Kovad, Varud, Chikhali is $5x^2 – 3y^2, 7y^2 + 2xy$ and $9x^2 + 4xy$ respectively.
At the beginning of the year $2017$ , $ x^2 + xy – y^2$ , $5xy$ and $3x^2 + xy$ persons from each of the three villages respectively went to another village for education,
then what is the remaining total population of these three villages ?
AnswerTotal population of villages at the end of 2016
$=\left(5 x^2-3 y^2\right)+\left(7 y^2+2 x y\right)+\left(9 x^2+4 x y\right)$
$=5 x^2+9 x^2-3 y^2+7 y^2+2 x y+4 x y$
$=14 x^2+4 y^2+6 x y \ldots \ldots \text { (i) }$
Total number of persons who went to other village at the beginning of 2017
$=\left(x^2+x y-y^2\right)+(5 x y)+\left(3 x^2+x y\right)$
$=x^2+3 x^2-y^2+x y+5 x y+x y$
$=4 x^2-y^2+7 x y \ldots \text { (ii) }$
Remaining total population of villages = Total population at the end of 2016 - total number of persons who went to other village at the beginning of 2017
$=14 x^2+4 y^2+6 x y-\left(4 x^2-y^2+7 x y\right) \ldots[\text { From (i) and (ii) }]$
$=14 x^2+4 y^2+6 x y-4 x^2+y^2-7 x y$
$=14 x^2-4 x^2+4 y^2+y^2+6 x y-7 x y=1$
$=10 x^2+5 y^2-x y$
$\therefore$ The remaining total population of the three villages is $10 x^2+5 y^2-x y$.
View full question & answer→Question 461 Mark
For which value of $m, x+3$ is the factor of the polynomial $x^3-2 m x+21$ ?
AnswerHere, $p(x)=x^3-2 m x+21$
$(x+3)$ is a factor of $x^3-2 m x+21$.
$\therefore$ By factor theorem,
Remainder $=0$
$\therefore P(-3)=0$
$p(x)=x^3-2 m x+21$
$\therefore p(-3)=(-3)^3-2(m)(-3)+21$
$\therefore 0=-27+6 m+21$
$\therefore 6+6 m=0$
$\therefore 6 m=6$
$\therefore m=1$
$\therefore x+3$ is the factor of $x ^3-2 mx +21$ for $m =1$.
View full question & answer→Question 471 Mark
Divide polynomial $3x^3 – 8x^2 + x + 7$ by $x – 3$ using synthetic method and write the quotient and remainder.
AnswerDividend = $3 x^3-8 x^2+x+7$
∴ Coefficient form of dividend = $(3, – 8, 1,7)$
Divisor =$ x – 3$
∴ Opposite of $– 3$ is $3$
Coefficient form of quotient = $(3, 1,4)$
∴ Quotient = $3x^2 + x + 4$ and
Remainder =$19$ View full question & answer→Question 481 Mark
Multiply the following polynomials.
i. $\left(m^3-2 m+3\right)\left(m^4-2 m^2+3 m+2\right)$
ii. $\left(5 m^3-2\right)\left(m^2-m+3\right)$
Answeri. $(m^3 – 2m + 3) (m^4 – 2m^2 + 3m + 2)$
$= m^3(m^4 – 2m^2 + 3m + 2) – 2m(m^4 – 2m^2 + 3m + 2) + 3(m^4 – 2m^2 + 3m + 2)$
$= m^7 – 2m^5 + 3m^4 + 2m^3 – 2m^5 + 4m^3 – 6m^2 – 4m + 3m^4 – 6m^2 + 9m + 6$
$= m^7 – 2m^5 – 2m^5 + 3m^4 + 3m^4 + 2m^3 + 4m^3 – 6m^2 – 6m^2 – 4m + 9m + 6$
$= m^7 – 4m^5 + 6m^4 + 6m^3 – 12m^2 + 5m + 6ii. (5m^3 – 2) (m^2 – m + 3)$
$= 5m^3(m^2 – m + 3) – 2(m^2 – m + 3)$
$= 5m^5 – 5m^4 + 15m^3 – 2m^2 + 2m – 6$
View full question & answer→Question 491 Mark
Subtract the second polynomial from the first.
i. $5 x^2-2 y+9 ; 3 x^2+5 y-7$
ii. $2 x^2+3 x+5 ; x^2-2 x+3$
Answeri. $\left(5 x^2-2 y+9\right)-\left(3 x^2+5 y-7\right)$
$=5 x^2-2 y+9-3 x^2-5 y+1$
$=5 x^2-3 x^2-2 y-5 y+9+7$
$=2 x^2-1 y+16 i i .\left(2 x^2+3 x+5\right)-\left(x^2-2 x+3\right)$
$=2 x^2+3 x+5-x^2+2 x-3$
$=2 x^2-x^2+3 x+2 x+5-3$
$=x^2+5 x+2$
View full question & answer→Question 501 Mark
Add the following polynomials.
i.$7 x^4-2 x^3+x+10$
$3 x^4+15 x^3+9 x^2-8 x+2$
ii. $3 p^3 q+2 p^2 q+7$
$2 p^2 q+4 p q-2 p^3 q$
Answeri. $\left(7 x^4-2 x^3+x+10\right)+\left(3 x^4+15 x^3+9 x^2-8 x+2\right)$
$=7 x^4-2 x^3+x+10+3 x^4+15 x^3+9 x^2-8 x+2$
$=7 x^4+3 x^4-2 x^3+15 x^3+9 x^2+x-8 x+10+2$
$=10 x^4+13 x^3+9 x^2-7 x+12 \text { ii. }\left(3 p^3 q+2 p^2 q+7\right)+\left(2 p^2 q+4 p q-2 p^3 q\right)$
$=3 p^3 q+2 p^2 q+7+2 p^2 q+4 p q-2 p^3 q$
$=3 p^3 q-2 p^3 q+2 p^2 q+2 p^2 q+4 p q+7$
$=p^3 q+4 p^2 q+4 p q+7$
View full question & answer→Question 511 Mark
Write the index form of the polynomial using variable x from its coefficient form.
i. $(3,-2,0,7,18)$
ii. $(6,1,0,7)$
iii. $(4,5,-3,0)$
Answeri. Number of coefficients $=5$
$\therefore$ Degree $=5-1=4$
$\therefore$ Index form $=3 x^4-2 x^3+0 x^2+7 x+18 i i$. Number of coefficients $=4$
$\therefore$ Degree $=4-1=3$
$\therefore$ Index form $=6 x^3+x^2+0 x+7$
iii. Number of coefficients $=4$
$\therefore$ Degree $=4-1=3$
$\therefore$ Index form $=4 x^3+5 x^2-3 x+0$
View full question & answer→Question 521 Mark
Write the following polynomial in coefficient form.
i. $x^4+16$
ii. $m^5+2 m^2+3 m+15$
Answeri. $x^4+16$
Index form $=x^4+0 x^3+0 x^2+0 x+16$
$\therefore$ Coefficient form of the polynomial $=(1,0,0,0,16)$ ii. $m ^5+2 m^2+3 m+15$
Index form $=m^5+0 m^4+0 m^3+2 m^2+3 m+15$
$\therefore$ Coefficient form of the polynomial $=(1,0,0,2,3,15)$
View full question & answer→Question 531 Mark
Write the following polynomials in standard form.
i. $4 x^2+7 x^4-x^3-x+9$
ii. $p+2 p^3+10 p^2+5 p^4-8$
Answeri. $7 x^4-x^3+4 x^2-x+9$
ii. $5 p^4+2 p^3+10 p^2+p-8$
View full question & answer→