Polynomials in One Variable Questions in English

(D) polynomial is classified based on its degree,which is the highest power of the variable present in the expression.
For the given polynomial $p(y) = y^{3}-y$,the highest power of the variable $y$ is $3$.
$A$ polynomial of degree $3$ is called a cubic polynomial.
Therefore,$y^{3}-y$ is a cubic polynomial.

(C) polynomial of degree $1$ is called a linear polynomial,a polynomial of degree $2$ is called a quadratic polynomial,and a polynomial of degree $3$ is called a cubic polynomial.
In the given expression $1+x+x^{2}$,the highest power of the variable $x$ is $2$.
Since the degree of the polynomial is $2$,it is classified as a quadratic polynomial.

(QUADRATIC) polynomial of degree $2$ is called a quadratic polynomial.
Since the degree of the polynomial $t^{2}$ is $2$,it is a quadratic polynomial.

(B) polynomial of degree $1$ is called a linear polynomial.
The given expression is $\sqrt{2} x - 1$.
Since the highest power of the variable $x$ is $1$,the degree of the polynomial is $1$.
Therefore,$\sqrt{2} x - 1$ is a linear polynomial.

(B) Let the polynomial be $p(x) = 3x^{3} - 4x^{2} + 7x - 5$.
To find the value of the polynomial at $x = 3$,we substitute $3$ for $x$ in the expression:
$p(3) = 3(3)^{3} - 4(3)^{2} + 7(3) - 5$
Calculate the powers:
$p(3) = 3(27) - 4(9) + 21 - 5$
Perform the multiplications:
$p(3) = 81 - 36 + 21 - 5$
Perform the addition and subtraction:
$p(3) = 45 + 21 - 5 = 66 - 5 = 61$.
Thus,the value of the polynomial at $x = 3$ is $61$.

(D) Given the polynomial $p(x) = x^2 - 4x + 3$.
First,calculate $p(2)$:
$p(2) = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1$.
Next,calculate $p(-1)$:
$p(-1) = (-1)^2 - 4(-1) + 3 = 1 + 4 + 3 = 8$.
Then,calculate $p\left(\frac{1}{2}\right)$:
$p\left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^2 - 4\left(\frac{1}{2}\right) + 3 = \frac{1}{4} - 2 + 3 = \frac{1}{4} + 1 = \frac{5}{4}$.
Finally,evaluate the expression $p(2) - p(-1) + p\left(\frac{1}{2}\right)$:
$= -1 - 8 + \frac{5}{4} = -9 + \frac{5}{4} = \frac{-36 + 5}{4} = \frac{-31}{4}$.

Given the polynomial $p(x) = 10x - 4x^{2} - 3$.
$(i)$ To find $p(0)$,substitute $x = 0$ into the polynomial:
$p(0) = 10(0) - 4(0)^{2} - 3 = 0 - 0 - 3 = -3$.
$(ii)$ To find $p(1)$,substitute $x = 1$ into the polynomial:
$p(1) = 10(1) - 4(1)^{2} - 3 = 10 - 4 - 3 = 3$.
$(iii)$ To find $p(-2)$,substitute $x = -2$ into the polynomial:
$p(-2) = 10(-2) - 4(-2)^{2} - 3 = -20 - 4(4) - 3 = -20 - 16 - 3 = -39$.

(C) According to the Remainder Theorem,if a polynomial $p(x)$ is divided by $(x - a)$,the remainder is $p(a)$.
Given $g(x) = x - 3$,we set $g(x) = 0$ to find the value of $x$:
$x - 3 = 0 \Rightarrow x = 3$.
Now,we substitute $x = 3$ into the polynomial $p(x)$:
$p(3) = (3)^{3} - 3(3)^{2} + 4(3) + 50$
$p(3) = 27 - 3(9) + 12 + 50$
$p(3) = 27 - 27 + 12 + 50$
$p(3) = 0 + 62 = 62$.
Therefore,the remainder is $62$.

(A) According to the Remainder Theorem,if a polynomial $p(x)$ is divided by $g(x) = ax + b$,the remainder is $p(-b/a)$.
Set $g(x) = 0$ to find the value of $x$:
$1 - \frac{3}{2}x = 0$
$\frac{3}{2}x = 1$
$x = \frac{2}{3}$
Now,substitute $x = \frac{2}{3}$ into $p(x)$:
$p\left(\frac{2}{3}\right) = \left(\frac{2}{3}\right)^{3} - 6\left(\frac{2}{3}\right)^{2} + 2\left(\frac{2}{3}\right) - 4$
$p\left(\frac{2}{3}\right) = \frac{8}{27} - 6\left(\frac{4}{9}\right) + \frac{4}{3} - 4$
$p\left(\frac{2}{3}\right) = \frac{8}{27} - \frac{24}{9} + \frac{4}{3} - 4$
To add these,find the common denominator,which is $27$:
$p\left(\frac{2}{3}\right) = \frac{8}{27} - \frac{72}{27} + \frac{36}{27} - \frac{108}{27}$
$p\left(\frac{2}{3}\right) = \frac{8 - 72 + 36 - 108}{27}$
$p\left(\frac{2}{3}\right) = \frac{-136}{27}$

(A) The given expression is $\pi x^{2}-\sqrt{3} x+11$.
$1$. $A$ polynomial is an algebraic expression in which the exponents of the variable are non-negative integers.
$2$. In the expression $\pi x^{2}-\sqrt{3} x+11$,the exponents of the variable $x$ are $2$,$1$,and $0$ (since $11 = 11x^{0}$).
$3$. Since all exponents are non-negative integers,the expression is a polynomial.
$4$. Since the expression contains only one variable,$x$,it is a polynomial in one variable.

(D) The given expression is $x^{2} + x - 3 + \frac{4}{x}$.
We can rewrite the term $\frac{4}{x}$ as $4x^{-1}$.
Thus,the expression becomes $x^{2} + x - 3 + 4x^{-1}$.
$A$ polynomial is defined as an algebraic expression where the exponents of the variables are non-negative integers.
In this expression,the exponent of $x$ in the term $4x^{-1}$ is $-1$,which is a negative integer.
Therefore,the given expression is not a polynomial.

(D) The given expression is $5 x^{2}-7 x+3 \sqrt{x}$.
To determine if an expression is a polynomial,the exponent of the variable must be a non-negative integer.
In the given expression,the term $3 \sqrt{x}$ can be written as $3 x^{\frac{1}{2}}$.
Here,the exponent of $x$ is $\frac{1}{2}$,which is not a non-negative integer.
Therefore,$5 x^{2}-7 x+3 \sqrt{x}$ is not a polynomial.

(N/A) The given expression $x^{3}+y^{3}+z^{3}-3xyz$ is a polynomial because the exponent of the variable in each of its terms is a non-negative integer (a whole number).
However,the expression contains three distinct variables: $x$,$y$,and $z$.
Therefore,it is a polynomial,but it is not a polynomial in one variable.

(D) The given expression is $x^{2}+2xy+y^{2}$.
$1$. $A$ polynomial is an algebraic expression in which the exponents of the variables are non-negative integers (whole numbers).
$2$. In the given expression,the exponents of $x$ and $y$ are $2, 1,$ and $0$,which are all whole numbers. Therefore,it is a polynomial.
$3$. However,the expression contains two distinct variables,$x$ and $y$.
$4$. Since it contains more than one variable,it is not a polynomial in one variable.

(A) The given expression is $x^{2}-8x+15$.
In this expression,the variable is $x$.
The exponents of $x$ in the terms $x^{2}$,$-8x^{1}$,and $15x^{0}$ are $2, 1$,and $0$,respectively.
Since all these exponents are non-negative integers,the expression is a polynomial.
Furthermore,since there is only one variable $x$ present in the expression,it is a polynomial in one variable.

(A) The given expression is $y^{2} + \sqrt{2}$.
This can be written as $y^{2} + \sqrt{2}y^{0}$.
$A$ polynomial in one variable is an algebraic expression where the exponent of the variable is a non-negative integer (a whole number).
In the expression $y^{2} + \sqrt{2}$,the exponent of the variable $y$ is $2$,which is a whole number.
Therefore,$y^{2} + \sqrt{2}$ is a polynomial in one variable.

(D) The given expression is $3 \sqrt{t} + t \sqrt{2}$.
This can be rewritten as $3 t^{1/2} + \sqrt{2} \cdot t$.
$A$ polynomial in one variable is an algebraic expression where the exponent of the variable must be a non-negative integer (a whole number).
In the term $3 t^{1/2}$,the exponent of the variable $t$ is $\frac{1}{2}$,which is not a whole number.
Therefore,$3 \sqrt{t} + t \sqrt{2}$ is not a polynomial.

(D) The given expression is $y + \frac{2}{y}$.
This can be rewritten as $y + 2 \cdot y^{-1}$.
$A$ polynomial in one variable is an algebraic expression where the exponent of the variable must be a non-negative integer (a whole number).
In the term $2 \cdot y^{-1}$,the exponent of the variable $y$ is $-1$.
Since $-1$ is not a whole number,the expression $y + \frac{2}{y}$ is not a polynomial.

(D) The given expression is $x^{10} + y^{3} + t^{50}$.
$1$. $A$ polynomial in one variable is an algebraic expression where the exponent of the variable is a non-negative integer and there is only one variable present.
$2$. In the expression $x^{10} + y^{3} + t^{50}$,there are three distinct variables: $x$,$y$,and $t$.
$3$. Although the exponent of each variable $(10, 3, 50)$ is a whole number,the expression contains three variables.
$4$. Therefore,$x^{10} + y^{3} + t^{50}$ is a polynomial in three variables,not in one variable.

(A) $(i)$ In the expression $\frac{\pi}{2} x^2 + x$,the term containing $x^2$ is $\frac{\pi}{2} x^2$. Therefore,the coefficient of $x^2$ is $\frac{\pi}{2}$.
$(ii)$ In the expression $\sqrt{2} x - 1$,there is no $x^2$ term. We can write this expression as $\sqrt{2} x - 1 + 0 \cdot x^2$. Therefore,the coefficient of $x^2$ is $0$.

(B) The degree of a polynomial is defined as the highest power of the variable present in the polynomial.
$(i)$ $5t - \sqrt{7}$
This is a polynomial in the variable $t$. The highest power of the variable $t$ is $1$. Therefore,the degree of this polynomial is $1$.
$(ii)$ $3$
This is a constant polynomial. The degree of any non-zero constant polynomial is always $0$,because $3$ can be written as $3 \times t^0$.

(N/A) $(i)$ $1+x$
Since the degree of $1+x$ is $1$,it is a linear polynomial.
$(ii)$ $3t$
Since the degree of $3t$ is $1$,it is a linear polynomial.
$(iii)$ $r^{2}$
Since the degree of $r^{2}$ is $2$,it is a quadratic polynomial.
$(iv)$ $7x^{3}$
Since the degree of $7x^{3}$ is $3$,it is a cubic polynomial.

$(16+\sqrt{11})$ Given the polynomial $q(y) = 3y^3 - 4y + \sqrt{11}$.
To find the value of the polynomial at $y = 2$,we substitute $2$ for $y$ in the expression:
$q(2) = 3(2)^3 - 4(2) + \sqrt{11}$
Calculate the power: $2^3 = 8$.
Substitute back: $q(2) = 3(8) - 8 + \sqrt{11}$.
Perform multiplication: $q(2) = 24 - 8 + \sqrt{11}$.
Perform subtraction: $q(2) = 16 + \sqrt{11}$.

(N/A) Given the polynomial $p(t) = 4t^4 + 5t^3 - t^2 + 6$.
To find the value of the polynomial at $t = a$,we substitute $a$ for $t$ in the expression.
$p(a) = 4(a)^4 + 5(a)^3 - (a)^2 + 6$.
Therefore,the value of the polynomial at $t = a$ is $4a^4 + 5a^3 - a^2 + 6$.

(B) Let the polynomial be $p(x) = 5x - 4x^2 + 3$.
To find the value at $x = -1$,substitute $-1$ for $x$ in the polynomial:
$p(-1) = 5(-1) - 4(-1)^2 + 3$
$p(-1) = -5 - 4(1) + 3$
$p(-1) = -5 - 4 + 3$
$p(-1) = -9 + 3$
$p(-1) = -6$

(C) Given polynomial is $p(x) = 5x - 4x^2 + 3$.
To find the value of the polynomial at $x = 2$,substitute $2$ for $x$ in the expression:
$p(2) = 5(2) - 4(2)^2 + 3$
$p(2) = 10 - 4(4) + 3$
$p(2) = 10 - 16 + 3$
$p(2) = -6 + 3$
$p(2) = -3$
Therefore,the value of the polynomial at $x = 2$ is $-3$.

(A) The given polynomial is $5-7 x-3 x^{2}$.
To find the coefficient of $x^{2}$,we look at the term containing $x^{2}$,which is $-3 x^{2}$.
The numerical factor multiplying $x^{2}$ is $-3$.
Therefore,the coefficient of $x^{2}$ is $-3$.

(A) The given polynomial is $\sqrt{3} x^{2}+11$.
To find the coefficient of $x^{2}$,we look at the term containing $x^{2}$.
In the expression $\sqrt{3} x^{2}+11$,the term with $x^{2}$ is $\sqrt{3} x^{2}$.
The numerical factor multiplying $x^{2}$ is $\sqrt{3}$.
Therefore,the coefficient of $x^{2}$ is $\sqrt{3}$.

(A) The given polynomial is $\pi x^{2}-\frac{22}{7} x+3.14$.
To find the coefficient of $x^{2}$,we look at the term containing $x^{2}$,which is $\pi x^{2}$.
The numerical factor multiplying $x^{2}$ is $\pi$.
Therefore,the coefficient of $x^{2}$ is $\pi$.

(0) In the polynomial $7 x^{3}-11 x+24$,the term containing $x^{2}$ is not present.
This can be written as $7 x^{3} + 0x^{2} - 11 x + 24$.
Therefore,the coefficient of $x^{2}$ is $0$.

(3) The degree of a polynomial is defined as the highest power of the variable present in the polynomial.
In the given polynomial $7 x^{3}-9 x^{2}+4 x-22$,the powers of the variable $x$ are $3, 2, 1,$ and $0$ (for the constant term).
The highest power among these is $3$.
Therefore,the degree of the polynomial is $3$.

(0) constant polynomial is a polynomial of degree $0$.
Since $5$ can be written as $5 \times x^{0}$,the exponent of the variable $x$ is $0$.
Therefore,the degree of the constant polynomial $5$ is $0$.

(2) The degree of a polynomial is defined as the highest power of the variable present in the expression.
In the given polynomial $11-2 y^{2}$,the variable is $y$.
The term $11$ can be written as $11y^{0}$,and the term $-2y^{2}$ has a power of $2$.
Comparing the powers $0$ and $2$,the highest power is $2$.
Therefore,the degree of the polynomial $11-2 y^{2}$ is $2$.

(1) The given polynomial is $p(t) = \sqrt{11} t + 14$.
In a polynomial,the degree is defined as the highest power of the variable present in the expression.
Here,the variable is $t$,and its exponent is $1$.
Therefore,the degree of the polynomial $\sqrt{11} t + 14$ is $1$.

(A) The degree of a polynomial is the highest power of the variable present in the expression.
For the polynomial $5t + 3$,the variable is $t$ and its highest power is $1$.
$A$ polynomial of degree $1$ is called a linear polynomial.
Therefore,$5t + 3$ is a linear polynomial.

(B) The degree of a polynomial is the highest power of the variable present in the expression.
For the given polynomial $x^{2}-9x+14$,the highest power of the variable $x$ is $2$.
$A$ polynomial of degree $2$ is called a quadratic polynomial.
Therefore,$x^{2}-9x+14$ is a quadratic polynomial.

(C) The degree of a polynomial is the highest power of the variable present in the expression.
For the polynomial $8x^{3} - 343$,the highest power of the variable $x$ is $3$.
Since the degree of the polynomial is $3$,it is classified as a cubic polynomial.

(A) The given expression is $x^{2}-15x+50$.
$1$. $A$ polynomial is an algebraic expression in which the exponents of the variables are non-negative integers.
$2$. In the expression $x^{2}-15x+50$,the variable is $x$. The exponents of $x$ are $2$ and $1$,which are non-negative integers.
$3$. Therefore,it is a polynomial.
$4$. Since the expression contains only one variable $x$,it is a polynomial in one variable.

(N/A) The expression $x^{2}+y^{2}+z^{2}+2xy+2yz+2zx$ is a polynomial because it consists of variables with non-negative integer exponents and coefficients that are real numbers. Since it contains three distinct variables $(x, y, z)$,it is a polynomial in three variables,not a polynomial in one variable.

(N/A) The given expression is $5x^2 + 11x - 2\sqrt{x}$.
In the term $-2\sqrt{x}$,the exponent of $x$ is $1/2$,which is not a non-negative integer.
Since a polynomial must have only non-negative integer exponents for its variables,this expression is not a polynomial.

(N/A) The given expression is $3x^2 + 5x - 7 + \frac{8}{x}$.
This can be rewritten as $3x^2 + 5x - 7 + 8x^{-1}$.
$A$ polynomial is an algebraic expression in which the exponents of the variables are non-negative integers.
In this expression,the term $8x^{-1}$ has an exponent of $-1$,which is a negative integer.
Therefore,the expression is not a polynomial.

(A) An expression is a polynomial if the exponents of the variable are non-negative integers. In the expression $\sqrt{3} x^{2}+\pi x-9$,the variable is $x$. The exponents of $x$ are $2$ and $1$,which are non-negative integers. Therefore,it is a polynomial. Since the expression contains only one variable $x$,it is a polynomial in one variable.

(B) The given polynomial is $p(x) = x^{3} + 27$.
To find the coefficient of $x^{2}$,we can rewrite the polynomial as $p(x) = 1 \cdot x^{3} + 0 \cdot x^{2} + 0 \cdot x + 27$.
Comparing this with the standard form,the term containing $x^{2}$ is $0 \cdot x^{2}$.
Therefore,the coefficient of $x^{2}$ is $0$.

(C) The given polynomial is $4+7x+3x^{2}$.
To find the coefficient of $x^{2}$,we look at the term containing $x^{2}$.
In the expression $4+7x+3x^{2}$,the term containing $x^{2}$ is $3x^{2}$.
The coefficient is the numerical factor multiplying the variable part.
Therefore,the coefficient of $x^{2}$ is $3$.

(N/A) The given polynomial is $p(x) = \sqrt{5}x^{2} - 7x + 13$.
The coefficient of $x^{2}$ is the numerical factor multiplying the $x^{2}$ term.
In this expression,the term containing $x^{2}$ is $\sqrt{5}x^{2}$.
Therefore,the coefficient of $x^{2}$ is $\sqrt{5}$.

(A) The given polynomial is $3x^{3} - 8x^{2} + 14x - 5$.
To find the coefficient of $x^{2}$,we look at the term containing $x^{2}$,which is $-8x^{2}$.
The numerical factor multiplying $x^{2}$ is $-8$.
Therefore,the coefficient of $x^{2}$ is $-8$.

(B) The degree of a polynomial is defined as the highest power of the variable present in the polynomial expression.
In the given polynomial $p(x) = x^{50} - 1$,the variable is $x$.
The powers of $x$ present are $50$ (in $x^{50}$) and $0$ (since $1 = 1 \cdot x^0$).
The highest power among these is $50$.
Therefore,the degree of the polynomial $x^{50} - 1$ is $50$.

(C) The degree of a polynomial is defined as the highest power of the variable present in the polynomial expression.
In the given polynomial $8x^{5} + 3x^{2} - 4x + 7$,the powers of the variable $x$ are $5, 2, 1,$ and $0$ (since $7 = 7x^{0}$).
The highest power among these is $5$.
Therefore,the degree of the polynomial is $5$.

(D) The given polynomial is $p(x) = x^{3} - 3(x^{2})^{4} - 15$.
First,simplify the term $(x^{2})^{4}$ using the power of a power rule $(a^{m})^{n} = a^{m \times n}$.
$(x^{2})^{4} = x^{2 \times 4} = x^{8}$.
Substituting this back into the polynomial,we get $p(x) = x^{3} - 3x^{8} - 15$.
The degree of a polynomial is defined as the highest power of the variable present in the expression.
In the expression $x^{3} - 3x^{8} - 15$,the powers of $x$ are $3$,$8$,and $0$ (since $15 = 15x^{0}$).
The highest power is $8$.
Therefore,the degree of the polynomial is $8$.

(A) The degree of a polynomial is defined as the highest power of the variable present in the expression.
In the given polynomial $5x^{2} + 12x + 4$,the powers of the variable $x$ are $2$,$1$,and $0$ (since $4 = 4x^{0}$).
The highest power among these is $2$.
Therefore,the degree of the polynomial $5x^{2} + 12x + 4$ is $2$.