Question 11 Mark
Identify constant, linear, quadratic, cubic and quadrtic polynomials from the following:
$-6 x^2$
Answer$-6 x^2$ is a polynomial with degree $2$. So, it is a quadratic polynomial.
View full question & answer→Question 21 Mark
Which of the following expressions are polynomials? In case of a polynomial, write its degree. $\text{x}^5-2\text{x}^3+\text{x}+\sqrt3$
Answer$\text{x}^5-2\text{x}^3+\text{x}+\sqrt3$ is an expression having only non-negative integral powers of $x$. So, it is a polynomial. Also, the highest power of $x$ is $5$, so, it is polynimial of degree $5.$
View full question & answer→Question 31 Mark
Write: The cofficient of x in $\sqrt3-2\sqrt2\text{x}+6\text{x}^2.$
AnswerThe cofficient of x in $\sqrt3-2\sqrt2\text{x}+6\text{x}^2$ is $-2\sqrt2.$
View full question & answer→Question 41 Mark
Which of the following expressions are polynomials? In case of a polynomial, write its degree. $\text{x}^{100}-1$
Answer$\text{x}^{100}-1$ is an expression having only non-negative integral powers of $x.$ So, it is a polynomial. Also, the highest power of $x$ is $100$, so, it is polynimial of degree $100.$
View full question & answer→Question 51 Mark
Identify constant, linear, quadratic, cubic and quadrtic polynomials from the following:
$-7 + x$
Answer$-7 + x$ is a polynomial with degree $1$. So, it is a linear polynomial.
View full question & answer→Question 61 Mark
Rewrite the following polynomial in standard form. $\frac{2}{3}+4\text{y}^2-3\text{y}+2\text{y}^3$
Answer$\frac{2}{3}-3\text{y}+4\text{y}^2+2\text{y}^3$ is a polynomial in standard form as the powers of $y$ are in ascending order.
View full question & answer→Question 71 Mark
Which of the following expressions are polynomials? In case of a polynomial, write its degree. $\frac{\text{x}^2}{2}-\frac{2}{\text{x}^2}$
Answer$\frac{\text{x}^2}{2}-\frac{2}{\text{x}^2}=\frac{\text{x}^2}{2}-2\text{x}^{-2}$ This is an expression having negative integral power of $x$ i.e. $-2$. So, it is not a polynomial.
View full question & answer→Question 81 Mark
Determine the degree of the following polynomials. $(3 x-2)\left(2 x^3+3 x^2\right)$
Answer$(3 x-2)\left(2 x^3+3 x^2\right)=6 x^4+9 x^3-4 x^3-6 x^2=6 x^4+5 x^3-6 x^2$
Here, the highest power of $x$ is $4 $. So, the degree of the polynomial is $4 .$
View full question & answer→Question 91 Mark
Which of the following expressions are polynomials? In case of a polynomial, write its degree.
$\text{x}^{-2}+2\text{x}^{-1}+3$
Answer$\text{x}^{-2}+2\text{x}^{-1}+3$ is an expression having negative integral powers of $x$. So, it is not a polynomial.
View full question & answer→Question 101 Mark
Find the zero of the polynomial:$q(x) = 4x$
Answer$q(x) = 0$
$\Rightarrow 4x = 0$
$\Rightarrow x = 0$
Hence, 0 is the zero of the polynomial $q(x).$
View full question & answer→Question 111 Mark
Identify constant, linear, quadratic, cubic and quadrtic polynomials from the following: $1-y-y^3$
Answer$1-y-y^3$ is a polynomial with degree $3$ . So, it is a cubic polynomial.
View full question & answer→Question 121 Mark
Find the zero of the polynomial: $f(x) = 3x + 1$
Answer$f(x) = 0 \Rightarrow 3x + 1 = 0 \Rightarrow\text{x}=-\frac{1}{3}$ Hence, $-\frac{1}{3}$ is the zero of the polynomial $f(x).$
View full question & answer→Question 131 Mark
Which of the following expressions are polynomials? In case of a polynomial, write its degree. $\text{t}^2-\frac{2}{5}\text{t}+\sqrt{5}$
Answer$\text{t}^2-\frac{2}{5}\text{t}+\sqrt{5}$ is an expression having only non-negative integral powers of t. So, it is a polynomial. Also, the highest power of t is $2$, so, it is polynimial of degree $2.$
View full question & answer→Question 141 Mark
Identify constant, linear, quadratic, cubic and quadrtic polynomials from the following: $6y$
Answer$6y$ is a polynomial with degree $1$. So, it is a linear polynomial.
View full question & answer→Question 151 Mark
Which of the following expressions are polynomials? In case of a polynomial, write its degree. $\frac{3}{5}\text{x}^2-\frac{7}{3}\text{x}+9$
Answer$\frac{3}{5}\text{x}^2-\frac{7}{3}\text{x}+9$ is an expression having only non-negative integral powers of $x$. So, it is a polynomial. Also, the highest power of x is $2$, so, it is a polynomial of degree $2.$
View full question & answer→Question 161 Mark
Verify that: $4$ is a zero of the polynomial, $p(x) = x - 4.$
Answer$p(x) = x - 4 \Rightarrow p(4) = 4 - 4 = 0$ Hence, $4$ is the zero of the given polynomial.
View full question & answer→Question 171 Mark
Write: The cofficient of $x^2$ in $2 x-3+x^3$.
Answer$2 x-3+x^3=-3+2 x+0 x^2+x^3$
The cofficient of $x^2$ in $2 x-3+x^3$ is $0.$
View full question & answer→Question 181 Mark
Verify that: $\frac{-1}{2}$ is a zero of the polynomial, $g(y) = 2y + 1.$
Answer$\text{p}(\text{y}) = 2\text{y}+ 1$ $\Rightarrow\text{p}\Big(-\frac{1}{2}\Big)=2\times\Big(-\frac{1}{2}\Big)+1$ $=-1+1$ $=0$ Hence, $-\frac{1}{2}$ is the zero of the given polynomial.
View full question & answer→Question 191 Mark
Write: The constant term in $\frac{\pi}{2}\text{x}^2+7\text{x}-\frac{2}{5}\pi.$
AnswerThe constant term in $\frac{\pi}{2}\text{x}^2+7\text{x}-\frac{2}{5}\pi$ is $-\frac{2}{5}\pi.$
View full question & answer→Question 201 Mark
Which of the following expressions are polynomials? In case of a polynomial, write its degree. $1$
AnswerClearly, $1$ is a constant polynomial of degree$ 0.$
View full question & answer→Question 211 Mark
Which of the following expressions are polynomials? In case of a polynomial, write its degree. $\sqrt[3]{2}\text{x}^2-8$
Answer$\sqrt[3]{2}\text{x}^2-8$ is an expression having only non-negative power of $x$. So, it is a polynomial. Also, the highest power of $x$ is $2$, so, it is a polynomial of degree $2.$
View full question & answer→Question 221 Mark
If $p(x)=x^3-5 x^2+4 x-3$ and $g(x)=x-2$, show that $p(x)$ is not a multiple of $g(x)$.
Answer$p(x)$ is a multiple of $g(x)$ or not $\because g(x)=x-2$ [given]
Then, zero of $g(x)$ is $2$ . Now, $p(2)=(2)^3-5(2)^2+4(2)-3\left[\because p(x)=x^3-5 x^2+4 x-3\right.$, given $]$
$=8-20+8-3=7 \neq 0$
Since, remainder $\neq 0$, so $p(x)$ is not a multiple of $g(x)$.
View full question & answer→Question 231 Mark
Identify constant, linear, quadratic, cubic and quadrtic polynomials from the following:$ -13$
Answer$-13$ is a polynomial with degree $0$. So, it is a constant polynomial.
View full question & answer→Question 241 Mark
Verify that: $\frac{2}{5}$ is a zero of the polynomial,$ f(x) = 2 - 5x.$
Answer$\text{f}(\text{x}) = 2 - 5\text{x}$ $\Rightarrow\Big(\frac{2}{5}\Big)=2-5\times\Big(\frac{2}{5}\Big)$ $=2-2$ $=0$ Hence, $\frac{2}{5}$ is the zero of the given polynomial.
View full question & answer→Question 251 Mark
Identify constant, linear, quadratic, cubic and quadrtic polynomials from the following:
$-z^3$
Answer$6y$ is a polynomial with degree $1$. So, it is a linear polynomial.
View full question & answer→Question 261 Mark
Rewrite the following polynomial in standard form.
$6 x^3+2 x-x^5-3 x^2$
Answer$2 x-3 x^2+6 x^3-x^5$ is a polynomial in standard form as the powers of x are in ascending order.
View full question & answer→Question 271 Mark
Determine the degree of the following polynomials. $\frac{4\text{x}-5\text{x}^2+6\text{x}^3}{2\text{x}}$
Answer$\frac{4\text{x}-5\text{x}^2+6\text{x}^3}{2\text{x}}=\frac{4\text{x}}{2\text{x}}-\frac{5\text{x}^2}{2\text{x}}+\frac{6\text{x}^3}{2\text{x}}=2-\frac{5}{2}\text{x}+3\text{x}^2$ Here, the highest power of $x$ is $2$. So, the degree of the polynomial is $2.$
View full question & answer→Question 281 Mark
Write:
The coefficient of $x^3$ in $x+3 x^2-5 x^3+x^4$.
AnswerThe cofficient of $x^3$ in $x+3 x^2-5 x^2+x^4$ is -5 .
View full question & answer→Question 291 Mark
Which of the following expressions are polynomials? In case of a polynomial, write its degree. $2\text{x}^3+3\text{x}^2+\sqrt{\text{x}}-1$
Answer$2\text{x}^3+3\text{x}^2+\sqrt{\text{x}}-1$ $=2\text{x}^3+3\text{x}^2+\text{x}^\frac{1}{2}-1$In this expression, one of the powers of x is $\frac{1}{2}$ which is a fraction. Since it is an expression having fractional power of $x$, so, it is not a polynomial.
View full question & answer→Question 301 Mark
Find the zero of the polynomial: $g(x) = 5 - 4x$
Answer$g(x) = 0 ⇒ 5 - 4x = 0$ $\Rightarrow\text{x}=\frac{5}{4}$ Hence, $\frac{5}{4}$ is the zero of the polynomial $g(x).$
View full question & answer→Question 311 Mark
Identify constant, linear, quadratic, cubic and quadrtic polynomials from the following: $-p$
Answer$-p$ is a polynomial with degree $1$. So, it is a linear polynomial.
View full question & answer→Question 321 Mark
Find the zero of the polynomial: $q(x) = x + 4$
Answer$q(x) = 0 \Rightarrow x + 4 = 0 \Rightarrow x = -4$ Hence, $-4$ is the zero of the polynomial $q(x).$
View full question & answer→Question 331 Mark
Find the zero of the polynomial: $p(x) = x - 5$
Answer$p(x) = 0 $
$\Rightarrow x - 5 = 0 $
$\Rightarrow x = 5$
Hence, $5$ is the zero of the polynomial $p(x).$
View full question & answer→Question 341 Mark
Which of the following expressions are polynomials? In case of a polynomial, write its degree. $\frac{-3}{5}$
AnswerClearly, $\frac{-3}{5}$ is a constant polynomial of degree $0.$
View full question & answer→Question 351 Mark
Which of the following expressions are polynomials? In case of a polynomial, write its degree. $\text{x}^4-\text{x}^\frac{3}{2}+\text{x}-3$
Answer$\text{x}^4-\text{x}^\frac{3}{2}+\text{x}-3$ In this expression, one of the powers of x is $\frac{3}{2}$ which is a fraction. Since it is an expression having fractional power of x, so, it is not a polynomial.
View full question & answer→Question 361 Mark
Rewrite the following polynomial in standard form.
$2+t-3 t^3+t^4-t^2$
Answer$2+t-t^2-3 t^3+t^4$ is a polynomial in standard form as the powers of $t$ are in ascending order.
View full question & answer→Question 371 Mark
Identify constant, linear, quadratic, cubic and quadrtic polynomials from the following:
$1+x+x^2$
Answer$1+x+x^2$ is a polynomial with degree $2$. So, it is a quadratic polynomial.
View full question & answer→Question 381 Mark
Which of the following expressions are polynomials? In case of a polynomial, write its degree. $\frac{1}{\sqrt2}\text{x}^2-\sqrt2\text{x}+2$
Answer$\frac{1}{\sqrt2}\text{x}^2-\sqrt2\text{x}+2$ is an expression having only non-negative integral powers of $x$. So, it is a polynomial. Also, the highest power of $x$ is $2$, so, it is polynimial of degree $2.$
View full question & answer→Question 391 Mark
Which of the following expressions are polynomials? In case of a polynomial, write its degree. $\frac{1}{2\text{x}^2}$
Answer$\frac{1}{2\text{x}^2}=\frac{1}{2}\text{x}^{-2}$ is an expression having negative power of $x$. So, it is not a polynomial.
View full question & answer→Question 401 Mark
Find the zero of the polynomial: $h(x) = 6x - 2$
Answer$h(x) = 0 \Rightarrow 6x - 1 = 0 $$\Rightarrow\text{x}=\frac{1}{6}$ Hence, $\frac{1}{6}$ is the zero of the polynomial $h(x).$
View full question & answer→Question 411 Mark
Determine the degree of the following polynomials.
$x^{-2}\left(x^4+x^2\right)$
Answer$x^{-2}\left(x^4+x^2\right)=x^2+x^0=x^2+1$
Here, the highest power of $x$ is $2$ . So, the degree of the polynomial is $2 .$
View full question & answer→Question 421 Mark
Verify that: $-3$ is a zero of the polynomial, $q(x) = x + 3.$
Answer$q(x) = (-3) + 3 \Rightarrow q(-3) = 0$ Hence, $3$ is the zero of the given polynomial.
View full question & answer→Question 431 Mark
Determine the degree of the following polynomials.$ -8$
Answer$-8 -8$ is a constant polynomial. So, the degree of the polynomial.
View full question & answer→Question 441 Mark
Which of the following expressions are polynomials? In case of a polynomial, write its degree. $\text{y}^3+\sqrt{3}\text{y}$
Answer$\text{y}^3+\sqrt{3}\text{y}$ is an expression having only non-negative integral powers of $y$. So, it is a polynomial. Also, the highest power of $y$ is $3$, so, it is polynimial of degree $3.$
View full question & answer→Question 451 Mark
Write: The cofficient of $x$ in $\frac{3}{8}\text{x}^2-\frac{2}{7}\text{x}+\frac{1}{6}.$
AnswerThe cofficient of $x$ in $\frac{3}{8}\text{x}^2-\frac{2}{7}\text{x}+\frac{1}{6}$ is $\frac{2}{7}.$
View full question & answer→Question 461 Mark
Determine the degree of the following polynomials. $-\frac{1}{2}\text{x}+3$
Answer$-\frac{1}{2}\text{x}+3$ Here, the highest power of $x$ is $1.$ So, the degree of the polynomial is $1.$
View full question & answer→Question 471 Mark
Identify constant, linear, quadratic, cubic and quadrtic polynomials from the following:
$x-x^3+x^4$
Answer$x-x^3+x^4$ is a polynomial with degree $4$. So, it is a quadratic polynomial.
View full question & answer→Question 481 Mark
Find the zero of the polynomial: $r(x) = 2x + 5$
Answer$r(x) = 0 \Rightarrow 2x + 5 \Rightarrow 2x + 5 = 0 \Rightarrow 2x = -5 \Rightarrow\text{x}=-\frac{ 5}{2}$
View full question & answer→Question 491 Mark
Which of the following expressions are polynomials? In case of a polynomial, write its degree. $\frac{1}{\sqrt5}\text{x}^\frac{1}{2}+1$
Answer$\frac{1}{\sqrt5}\text{x}^\frac{1}{2}+1$ In this expression, the power of x is $\frac{1}{2}$ which is a fraction. Since it is an expression having fractional power of $x$, so, it is not a polynomial.
View full question & answer→Question 501 Mark
If $p(x)=2 x^3-11 x^2-4 x+5$ and $g(x)=2 x+1$, show that $p(x)$ is not a factor of $g(x).$
Answer$p(x)=2 x^3-11 x^2-4 x+1$
If $p(x)$ is divided by $(2x + 1)$, then $\text{f}\Big(\frac{-1}{2}\Big)$ is the remainder.
$\text{p}\Big(\frac{-1}{2}\Big)=2\Big(\frac{-1}{2}\Big)^3-11\Big(\frac{-1}{2}\Big)^2-4\Big(\frac{-1}{2}\Big)+1$
$=2\Big(\frac{-1}{8}\Big)-11\Big(\frac{1}{4}\Big)-4\Big(\frac{-1}{2}\Big)+1$
$=\frac{-1}{4}-\frac{11}{4}+2+1$
$= 3 - \frac{12}{4}$
$= 3-3$
$= 0$
$(2x + 1)$ is a factor of $g(x)$ as remainder is zero.
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