Mix Examples - Polynomials Questions in English

(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 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 $x^{3}+3 x^{2} y+3 x y^{2}+y^{3}$ is a polynomial because all the exponents of the variables $x$ and $y$ are non-negative integers. Since it contains two distinct variables,$x$ and $y$,it is a polynomial in two variables,not 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 $2$.

(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 $5a^{2} - 11a + 7$.

(N/A) Let $p(x) = x^{2}-x-6$.
For $x = 3$:
$p(3) = (3)^{2} - 3 - 6 = 9 - 3 - 6 = 0$.
Since $p(3) = 0$,$3$ is a zero of the polynomial $x^{2}-x-6$.
For $x = 5$:
$p(5) = (5)^{2} - 5 - 6 = 25 - 5 - 6 = 14$.
Since $p(5) \neq 0$,$5$ is not a zero of the polynomial $x^{2}-x-6$.

(B) Given polynomial is $p(x)=5 x-8$.
To find the zero of the polynomial,we set $p(x)=0$.
So,$5 x-8=0$.
Adding $8$ to both sides,we get $5 x=8$.
Dividing both sides by $5$,we get $x=\frac{8}{5}$.
Therefore,the zero of the polynomial $p(x)=5 x-8$ is $\frac{8}{5}$.

(C) To find the value of the polynomial $p(x) = x^{2}-7x+12$ at $x=1$,we substitute $1$ for $x$ in the expression.
$p(1) = (1)^{2} - 7(1) + 12$
$p(1) = 1 - 7 + 12$
$p(1) = -6 + 12$
$p(1) = 6$
Therefore,the value of the polynomial at $x=1$ is $6$.

(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 $x = 3$,we substitute $3$ for $x$ in the expression:
$p(3) = (3)^{2} - 7(3) + 12$
$p(3) = 9 - 21 + 12$
$p(3) = 21 - 21$
$p(3) = 0$
Therefore,the value of the polynomial at $x = 3$ is $0$.

(B) To find the value of the polynomial $p(x) = x^{2}-7x+12$ at $x=4$,we substitute $4$ for $x$ in the expression.
$p(4) = (4)^{2} - 7(4) + 12$
$p(4) = 16 - 28 + 12$
$p(4) = 28 - 28$
$p(4) = 0$
Therefore,the value of the polynomial at $x=4$ is $0$.

(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:
$\frac{35}{4} = 8 \frac{3}{4}$

To find the values,we substitute $x = 1$,$x = 2$,and $x = 4$ into the polynomial $p(x) = x^{3} - 7x^{2} + 14x - 8$.
$1$. For $p(1)$:
$p(1) = (1)^{3} - 7(1)^{2} + 14(1) - 8 = 1 - 7 + 14 - 8 = 0$.
$2$. For $p(2)$:
$p(2) = (2)^{3} - 7(2)^{2} + 14(2) - 8 = 8 - 7(4) + 28 - 8 = 8 - 28 + 28 - 8 = 0$.
$3$. For $p(4)$:
$p(4) = (4)^{3} - 7(4)^{2} + 14(4) - 8 = 64 - 7(16) + 56 - 8 = 64 - 112 + 56 - 8 = 0$.
Thus,$p(1) = 0$,$p(2) = 0$,and $p(4) = 0$.

(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 $t = 4$: $p(4) = (4)^{2} - 6(4) + 8 = 16 - 24 + 8 = 0$.

(N/A) To find the values of the polynomial $p(x) = x^{2} - 3x + 2$ at $x = 1, 2, 4$,we substitute the values into the expression:
$1$. For $x = 1$: $p(1) = (1)^{2} - 3(1) + 2 = 1 - 3 + 2 = 0$.
$2$. For $x = 2$: $p(2) = (2)^{2} - 3(2) + 2 = 4 - 6 + 2 = 0$.
$3$. For $x = 4$: $p(4) = (4)^{2} - 3(4) + 2 = 16 - 12 + 2 = 6$.
Thus,the values are $p(1) = 0$,$p(2) = 0$,and $p(4) = 6$.

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 $x = 4$:
$p(4) = (4)^3 + 9(4)^2 + 23(4) + 15 = 64 + 9(16) + 92 + 15 = 64 + 144 + 92 + 15 = 315$.