Exercises: Algebraic expressions

Identify the coefficient, the variable part, and the degree of the following monomial: $M(x,y) = -\frac{2}{3}x^4y^2$.

Evaluate the polynomial $P(x) = 3x^2 - 5x + 2$ for $x = -2$.

Write the following statement as an algebraic expression: "The cube of a number minus the square of a different number."

State which of the following algebraic expressions are polynomials and justify your answer:

• $A(x) = 3x^2 - \frac{5}{x}$

• $B(x) = \sqrt{2}x^3 - x$

• $C(x) = 4x^{1/2} + 1$

Combine the following like terms and simplify the result: $3x^2y - 5x^2y + 8x^2y$.

Multiply the following monomials: $(-3a^2b^3) \cdot (4ab^4)$.

Determine the value of $k$ so that the value of the polynomial $P(x) = 2x^3 - kx + 4$ at $x = 1$ is equal to $5$.

Write the algebraic expression that represents the perimeter of a rectangle whose base measures $3x$ and whose height measures $x + 2$.

A clothing store offers a $20\%$ discount on all its items. If you buy $x$ shirts that originally cost $\$25$ each and $y$ pairs of pants that cost $\$40$ each, write the algebraic expression for the total cost of the purchase. Calculate the cost if you buy $3$ shirts and $2$ pairs of pants.

The kinetic energy of a moving body is given by the expression $E_c = \frac{1}{2}mv^2$, where $m$ is the mass in pounds and $v$ is the speed in feet per second. Calculate the kinetic energy of a vehicle of $800$ lb traveling at $20$ ft/s.

The figure shows a square of side $2x$ containing a circle of radius $x$ inside it. Write the algebraic expression that represents the area of the shaded region (the area of the square outside the circle).

Evaluate the algebraic expression $P(x,y) = \frac{x^2 - 2xy + y^2}{x - y}$ for $x = \frac{1}{2}$ and $y = -\frac{3}{2}$. Hint: simplify the expression before substituting the values.

A phone company charges a fixed monthly fee of $\$15$, plus $\$0.12$ for each minute of call $m$ and $\$0.05$ for each megabyte consumed $d$. Write the algebraic expression for the monthly bill and calculate the total for a month in which $100$ minutes and $500$ megabytes have been used.

Find the values of the parameters $a$ and $b$ in the polynomial $P(x) = ax^2 + bx - 5$ knowing that its value for $x = 1$ is $-2$ and for $x = -1$ is $-4$.

A cylindrical tank has a base radius $r = x$ and a height $h = 3x$. Write the algebraic expressions that represent its volume and its lateral surface area in terms of $x$.

Perform the following division of monomials and indicate the coefficient, the variable part, and the degree of the resulting monomial: $\frac{-24x^5y^4z}{6x^2y^3}$.

A box without a lid is to be made from a rectangular sheet of cardboard $40$ in long by $30$ in wide. To do this, squares of side $x$ are cut out from each of the four corners and the flaps are folded up. Write the algebraic expression for the volume of the box in terms of $x$ and calculate this volume if $x = 5$ in.

Prove algebraically that the difference between the squares of two consecutive odd numbers is always a multiple of $8$. (Remember that an odd number can be expressed in the form $2n+1$).

Given the algebraic expression $E(a,b) = \frac{a^3 - b^3}{a^2 + ab + b^2}$, fully simplify it by carrying out the division and use the result to mentally calculate its value when $a = 4.5$ and $b = 1.5$.

In an electrical circuit, the equivalent resistance $R_{eq}$ of three resistors $R_1$, $R_2$, and $R_3$ connected in parallel is obtained from the formula $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$. Solve for $R_{eq}$ and write its algebraic expression as a single fraction in terms of $R_1$, $R_2$, and $R_3$.

A rectangular garden has dimensions of $a$ feet long by $b$ feet wide. Two perpendicular paths of width $x$ are built parallel to the sides of the garden, forming a cross. Write the factored algebraic expression for the area of the remaining green zone.

Two car rental agencies offer different rates. Agency A charges a fixed fee of $\$30$ plus $\$0.20$ per mile driven. Agency B charges a fixed $\$50$ plus $\$0.15$ per mile. Write the algebraic expression that represents the difference in cost between Agency A and Agency B for a trip of $k$ miles, and determine for how many miles the cost is exactly the same.

Write the algebraic expression that gives the sum of the first $n$ positive integer multiples of $3$, knowing that the sum of the first $n$ natural numbers is $\frac{n(n+1)}{2}$. Use this expression to calculate the sum of the first $50$ multiples of $3$.

A farmer has $L$ feet of fencing to enclose a rectangular plot of land adjacent to a straight river. As the river acts as a natural barrier, they do not need to put a fence on that side. If we call $x$ the width of the plot (the dimension perpendicular to the river), write the algebraic expression for the area of the enclosure in terms of $L$ and $x$. What is the area if they have $100$ feet of fencing and the width is $20$ feet?