Introduction to Polynomials

If $p(x) = 7x - 2$, then $p(0)$ is:

What is the degree of the polynomial $y^{100} - 1$?

Find the value of $p(x) = x^4$ at $x = -1$.

The exponent of the variable in any term of a polynomial must be a:

Evaluate $104 \times 96$ using the identity $(a+b)(a-b) = a^2 - b^2$.

If $x + 1$ is a factor of $x^n + 1$, then $n$ must be:

Factorize $x^2 - \frac{y^2}{100}$.

If $p(x) = 5x - 4x^2 + 3$, find the value of $p(-1)$.

Factorize the polynomial $2x^2 - 18$.

Factorize the expression $a^6 - b^6$.

A monthly gym membership costs $₹1000$ plus $₹50$ for every personal training session. What is the cost for $s$ sessions?

A car has $40$ liters of fuel and uses $2$ liters per km. How much fuel is left after $x$ km?

A rectangle has a perimeter of $20$ cm. If the length is $x$ cm, what is the linear expression for its width?

If a shirt costs $s$ rupees and a discount of $₹100$ is given, what is the final price?

A concert organizer charges ₹1,200 per ticket and has to pay ₹20,000 for the venue. Find the linear polynomial for the profit if $x$ tickets are sold.

A person pays for $x$ pens costing ₹8 each with a ₹100 note. Find the linear polynomial representing the change received.

A square garden has a side length of $s$. If a 3-meter wide path is removed from one side of the perimeter, find the linear polynomial for the remaining length of the boundary.

A company’s profit (P(n)) in dollars for selling (n) units is given by (P(n) = 25n - 1500). If the cost to produce each unit is $10, find the linear polynomial representing the total revenue (R(n)) from selling (n) units. (Assume Revenue = Profit + Cost).

A gardener charges $30 for a visit plus $25 per hour. If the total bill was $117.50, find the linear polynomial (C(h)) for the cost and solve for the time (h) spent by the gardener.

Two cities are 300 km apart. A car leaves City A at 60 km/h. At the same time, another car leaves City B at 40 km/h traveling towards City A. Find the linear polynomial (D(t)) representing the distance between the two cars after (t) hours. At what time (t) will they meet?

Find the constant $c$ in the rule $y = 3x + c$ if $y = 7$ when $x = 2$.

What is the next term in the sequence $1, 1.1, 1.2, 1.3, \dots$?

If $y = x/2$, what is $y$ when $x=10$?

What is the $n^{th}$ term for the sequence $6, 12, 18, 24, \dots$?

The height of a stack of $n$ identical cookies is given by $H(n) = 0.8n + 0.2$ cm. What is the height of one cookie and the thickness of the base of the container?

Find the $n$-th term for a sequence where the first term is $2$ and the second term is $2.5$.

The values in a table are $(2, 5), (4, 9), (6, 13)$. What is the linear equation for this data?

If $f(x)$ is a linear polynomial and $f(1) = 4, f(2) = 1, f(3) = -2$, what is the sum of the coefficients of $f(x)$?

A sequence of figures is made of blocks. Figure 1 has 10 blocks, Figure 2 has 17 blocks, Figure 3 has 24 blocks. How many blocks will be in Figure $n$?

In a science experiment, the volume of a gas $V$ at constant pressure is a linear polynomial of its temperature $T$ ($V = kT + b$). If $V = 100$ at $T = 20$ and $V = 110$ at $T = 45$, find the value of $k$.

A shop sells juice. A $1000$ ml bottle is being used to fill cups of $200$ ml each. Find the polynomial $J(n)$ for the volume left in the bottle after $n$ cups are filled.

A cooling system reduces the temperature by $3^\circ\text{C}$ per hour. If the starting temperature is $35^\circ\text{C}$, find the polynomial $T(h)$ for the temperature after $h$ hours.

A person walks at a constant speed of $4$ km/h. If they have already walked $2$ km, find the polynomial $D(h)$ for the total distance after $h$ additional hours.

A forest has $1000$ trees. Due to a disease, $20$ trees die each year. Find the polynomial $T(y)$ for the number of trees left after $y$ years.

A bank account earns simple interest of ₹$50$ every month on a principal of ₹$2000$. Which polynomial represents the total amount $A(m)$ in the account after $m$ months?

The cost of a phone call is a connection fee of ₹$1.50$ plus ₹$0.10$ per minute. Which polynomial $C(m)$ represents the cost of an $m$-minute call?

A pool is filled with $5000$ gallons of water. It is being drained at $125$ gallons per hour. How many hours will pass until $2500$ gallons remain?

A worker is paid $₹400$ per day for a normal $8$-hour shift and $₹75$ per hour for overtime. If he earned $₹3325$ in a $6$-day week, how many total overtime hours did he work?

An oil tank is $2/5$ full. After adding $450$ liters, it becomes $7/8$ full. How many more liters are needed to fill the tank completely?

A courier service charges $₹40$ for the first $500$ grams and $₹15$ for each additional $250$ grams. A parcel weighs $2.75$ kg. Calculate the total shipping cost.

What is the slope of the line $y = x + 100$?

In the polynomial $p(x) = -7x$, what is the value of $b$?

If a line has a slope of $0$ and a y-intercept of $3$, its equation is:

What is the slope of the line $y = \frac{x}{3} + 7$?

A line is given by $y = ax + 3$. If it passes through $(4, 1)$, what is the value of $a$?

If $y = ax + b$ has $a = -5$ and $b = 10$, what is the x-coordinate when $y = 0$?

What is the slope of a line that passes through the origin and the point $(-3, -6)$?

If the linear polynomial $f(x) = ax + b$ satisfies $f(1) = 4$ and $f(x+2) = f(x) + 6$, find $f(0)$.

The distance between the $x$-intercept and $y$-intercept of the line $y = \frac{3}{4}x + 3$ is:

If the graph of $y = (k-1)x + k+1$ is parallel to the $x$-axis, find the $y$-intercept.