Polynomials in One Variable Questions in English

(B) The degree of a polynomial is defined as the highest power of the variable present in the polynomial expression.
In the given polynomial $ax^3 + bx^2 + cx + d$,the variable is $x$.
The powers of $x$ in the terms are $3, 2, 1,$ and $0$ (since $d = dx^0$).
The highest power among these is $3$.
Therefore,the degree of the polynomial is $3$.

(C) The degree of a polynomial is defined as the highest power of the variable present in the polynomial expression.
In the given polynomial $p(x) = x^{8} - 6561$,the variable is $x$.
The highest exponent of the variable $x$ is $8$.
Therefore,the degree of the polynomial $x^{8} - 6561$ is $8$.

(C) polynomial is classified based on its degree (the highest power of the variable).
For the given polynomial $p(x) = x^{3} + 5x^{2} + 12$,the highest power of the variable $x$ is $3$.
$A$ polynomial of degree $3$ is known as a cubic polynomial.
Therefore,$x^{3} + 5x^{2} + 12$ is a cubic polynomial.

(B) polynomial is classified based on its degree (the highest power of the variable).
For the given polynomial $4x^{2} - 49$,the highest power of the variable $x$ is $2$.
$A$ polynomial of degree $2$ is called a quadratic polynomial.
Therefore,$4x^{2} - 49$ is a quadratic polynomial.

(A) polynomial is classified based on its degree (the highest power of the variable).
$1$. The given expression is $5 - 3t$.
$2$. The variable in this expression is $t$.
$3$. The highest power of the variable $t$ is $1$.
$4$. $A$ polynomial of degree $1$ is called a linear polynomial.
Therefore,$5 - 3t$ is a linear polynomial.

(A) polynomial is classified based on its degree (the highest power of the variable).
For the given expression $4y + 11$,the variable is $y$ and its highest power is $1$.
$A$ polynomial of degree $1$ is called a linear polynomial.
Therefore,$4y + 11$ is a linear polynomial.

(C) polynomial is classified based on its degree (the highest power of the variable).
For the given polynomial $p(x) = x^{3}+2x^{2}+3x+2$,the highest power of the variable $x$ is $3$.
$A$ polynomial of degree $3$ is called a cubic polynomial.
Therefore,the given expression is a cubic polynomial.

(B) polynomial is classified based on its degree (the highest power of the variable).
$1$. $A$ polynomial of degree $1$ is called a linear polynomial.
$2$. $A$ polynomial of degree $2$ is called a quadratic polynomial.
$3$. $A$ polynomial of degree $3$ is called a cubic polynomial.
In the given expression $35x^{2} - 16x - 12$,the highest power of the variable $x$ is $2$.
Since the degree of the polynomial is $2$,it is a quadratic polynomial.

(B) Given polynomial is $p(x) = x^{2} + 5x - 24$.
To find the value of the polynomial at $x = 3$,we substitute $3$ for $x$ in the expression:
$p(3) = (3)^{2} + 5(3) - 24$
$= 9 + 15 - 24$
$= 24 - 24$
$= 0$
Therefore,the value of $p(x)$ at $x = 3$ is $0$.

(N/A) $q(y) = 5y^3 - 4y^2 + 14y - \sqrt{3}$
Replacing $y$ with $2$,we get:
$q(2) = 5(2)^3 - 4(2)^2 + 14(2) - \sqrt{3}$
$= 5(8) - 4(4) + 28 - \sqrt{3}$
$= 40 - 16 + 28 - \sqrt{3}$
$= 24 + 28 - \sqrt{3}$
$= 52 - \sqrt{3}$
Therefore,the value of $q(y)$ at $y = 2$ is $52 - \sqrt{3}$.

(N/A) Given the polynomial $p(t) = 5t^{2} - 11t + 7$.
To find the value of the polynomial at $t = a$,substitute $a$ in place of $t$ in the given expression.
$p(a) = 5(a)^{2} - 11(a) + 7$
$p(a) = 5a^{2} - 11a + 7$
Therefore,the value of the polynomial $p(t)$ at $t = a$ is $5a^{2} - 11a + 7$.

(D) Let the polynomial be $p(x) = x^{2}-7x+12$.
To find the value of the polynomial at $x=-2$,substitute $-2$ for $x$ in the expression:
$p(-2) = (-2)^{2} - 7(-2) + 12$
$p(-2) = 4 + 14 + 12$
$p(-2) = 30$.
Therefore,the value of the polynomial at $x=-2$ is $30$.

(A) To find the value of the polynomial $p(x) = x^{2} - 7x + 12$ at $x = \frac{1}{2}$,we substitute $\frac{1}{2}$ for $x$ in the expression:
$p(\frac{1}{2}) = (\frac{1}{2})^{2} - 7(\frac{1}{2}) + 12$
$p(\frac{1}{2}) = \frac{1}{4} - \frac{7}{2} + 12$
To add these,find a common denominator,which is $4$:
$p(\frac{1}{2}) = \frac{1}{4} - \frac{14}{4} + \frac{48}{4}$
$p(\frac{1}{2}) = \frac{1 - 14 + 48}{4}$
$p(\frac{1}{2}) = \frac{35}{4}$
Converting the improper fraction to a mixed number:
$\frac{35}{4} = 8 \frac{3}{4}$

(N/A) To find the values of the polynomial $p(y) = y^{2} - 5y + 4$ at given points,we substitute the value of $y$ into the expression:
$1$. For $y = 1$: $p(1) = (1)^{2} - 5(1) + 4 = 1 - 5 + 4 = 0$.
$2$. For $y = 2$: $p(2) = (2)^{2} - 5(2) + 4 = 4 - 10 + 4 = -2$.
$3$. For $y = 4$: $p(4) = (4)^{2} - 5(4) + 4 = 16 - 20 + 4 = 0$.
Thus,the values are $p(1) = 0$,$p(2) = -2$,and $p(4) = 0$.

(N/A) To find the values,substitute the given values of $t$ into the polynomial $p(t) = t^{2} - 6t + 8$.
For $t = 1$: $p(1) = (1)^{2} - 6(1) + 8 = 1 - 6 + 8 = 3$.
For $t = 2$: $p(2) = (2)^{2} - 6(2) + 8 = 4 - 12 + 8 = 0$.
For $t = 4$: $p(4) = (4)^{2} - 6(4) + 8 = 16 - 24 + 8 = 0$.

To find the values,substitute the given values of $x$ into the polynomial $p(x) = x^3 + 9x^2 + 23x + 15$:
$1$. For $x = 1$:
$p(1) = (1)^3 + 9(1)^2 + 23(1) + 15 = 1 + 9 + 23 + 15 = 48$.
$2$. For $x = 2$:
$p(2) = (2)^3 + 9(2)^2 + 23(2) + 15 = 8 + 9(4) + 46 + 15 = 8 + 36 + 46 + 15 = 105$.
$3$. For $x = 4$:
$p(4) = (4)^3 + 9(4)^2 + 23(4) + 15 = 64 + 9(16) + 92 + 15 = 64 + 144 + 92 + 15 = 315$.

(B) 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 $p(1/2)$:
$p(1/2) = (1/2)^{2} - 4(1/2) + 3 = 1/4 - 2 + 3 = 1/4 + 1 = 5/4$.
Finally,substitute these values into the expression $p(2) - p(-1) + p(1/2)$:
$-1 - 8 + 5/4 = -9 + 5/4 = (-36 + 5) / 4 = -31/4$.

(C) According to the remainder theorem,if a polynomial $p(x)$ is divided by $(x - a)$,the remainder is $p(a)$.
Here,the divisor is $x + 1$,which can be written as $x - (-1)$.
Therefore,we need to find $p(-1)$.
Given $p(x) = x^3 + x^2 - 26x + 24$.
Substitute $x = -1$ into the polynomial:
$p(-1) = (-1)^3 + (-1)^2 - 26(-1) + 24$
$p(-1) = -1 + 1 + 26 + 24$
$p(-1) = 0 + 50$
$p(-1) = 50$.
Thus,the remainder is $50$.

(B) The given polynomial is $p(x) = x^{2}-5x+4$.
$A$ polynomial is called a linear polynomial if its degree (the highest power of the variable) is $1$.
In the given expression $x^{2}-5x+4$,the highest power of the variable $x$ is $2$.
Since the degree of the polynomial is $2$,it is a quadratic polynomial,not a linear polynomial.
Therefore,the statement is False.

(B) The degree of a polynomial is defined as the highest power of the variable present in the expression.
In the polynomial $5x^{2} - 7x + 2$,the powers of the variable $x$ are $2$,$1$,and $0$ (since $2 = 2x^{0}$).
The highest power among these is $2$.
Therefore,the degree of the polynomial $5x^{2} - 7x + 2$ is $2$,not $5$.
Hence,the given statement is False.

(B) The given statement is False.
In the polynomial $5x^{3} - 3x^{2} + 11x - 14$,the term containing $x^{3}$ is $5x^{3}$.
The coefficient of $x^{3}$ is the numerical factor multiplying $x^{3}$,which is $5$,not $3$.

(B) polynomial is classified based on its degree (the highest power of the variable).
For the expression $4 x^{2}+11 x-3$,the highest power of the variable $x$ is $2$.
$A$ polynomial of degree $2$ is called a quadratic polynomial.
Therefore,$4 x^{2}+11 x-3$ is a quadratic polynomial.

(A) The degree of a polynomial is defined as the highest power of the variable present in the expression.
In the given polynomial $5x^{2} - 7x - 11$,the powers of the variable $x$ are $2$,$1$,and $0$ (since $11 = 11x^{0}$).
The highest power among these is $2$.
Therefore,the degree of the polynomial is $2$.

(A) The given polynomial is $p(x) = 5x^{3} - 3x^{2} + 11x - 4$.
To find the coefficient of $x^{2}$,we look at the term containing $x^{2}$,which is $-3x^{2}$.
The numerical factor multiplying $x^{2}$ is $-3$.
Therefore,the coefficient of $x^{2}$ is $-3$.

(A) Given the polynomial $p(x) = x^{3} - 3x^{2} + 8x + 12$.
To find $p(-1)$,substitute $x = -1$ into the polynomial:
$p(-1) = (-1)^{3} - 3(-1)^{2} + 8(-1) + 12$
$p(-1) = -1 - 3(1) - 8 + 12$
$p(-1) = -1 - 3 - 8 + 12$
$p(-1) = -12 + 12$
$p(-1) = 0$
Thus,the correct option is $A$.

(C) To find the degree of the polynomial,we first simplify the expression:
$7x^5 - 4x^4 + 2(x^3)^2 - x^2 + 35$
$= 7x^5 - 4x^4 + 2x^6 - x^2 + 35$
The degree of a polynomial is the highest power of the variable $x$ present in the expression.
In the simplified expression $2x^6 + 7x^5 - 4x^4 - x^2 + 35$,the highest power of $x$ is $6$.
Therefore,the degree of the polynomial is $6$.

(A) To find the value of $p(2)$,substitute $x = 2$ into the given polynomial $p(x) = x^{2} - 7x + 12$.
$p(2) = (2)^{2} - 7(2) + 12$
$p(2) = 4 - 14 + 12$
$p(2) = 16 - 14$
$p(2) = 2$
Therefore,the correct option is $A$.

(B) To find $p(-2)$,substitute $x = -2$ into the given polynomial $p(x) = 5x^2 - 11x + 3$.
$p(-2) = 5(-2)^2 - 11(-2) + 3$
$p(-2) = 5(4) + 22 + 3$
$p(-2) = 20 + 22 + 3$
$p(-2) = 45$
Therefore,the value of $p(-2)$ is $45$.