Operations with Polynomials
The robotics club at your school is selling hoodies. Making $x$ hoodies costs $C(x) = 15x + 300$ dollars, and the club lowers the price as it sells more: each hoodie sells for $p(x) = 50 - 0.5x$ dollars. Revenue, cost and profit are all polynomials, and profit is a new polynomial built from the other two. How do you combine them, and how many hoodies must the club sell just to break even? Operating with polynomials is the tool that turns several formulas into one you can evaluate, divide or factor.
Evaluating a polynomial
A polynomial is a sum of terms like $ax^n$, where $n$ is a whole number $0, 1, 2, \dots$. Its degree is the highest exponent, and standard form lists the terms from highest to lowest degree. To evaluate $P(x)$ at a number, substitute the number for every $x$, using parentheses, and then follow the order of operations.
Find the value of the polynomial $P(x) = -2x^3 + 5x^2 - 3x + 4$ for $x = -2$ and for $x = 1$.
Find the value of the polynomial $P(x) = -2x^3 + 5x^2 - 3x + 4$ for $x = -2$ and for $x = 1$.
Adding and subtracting polynomials
To add or subtract polynomials, combine like terms: terms with the same power of $x$. Lining them up in columns makes this easy. A number in front of a polynomial, like $3A(x)$, multiplies every term.
Given the polynomials $A(x) = 3x^2 - 5x + 1$ and $B(x) = -x^2 + 4x - 6$, find the resulting expression from performing the operation $2 \cdot A(x) - B(x)$.
Given the polynomials $A(x) = 3x^2 - 5x + 1$ and $B(x) = -x^2 + 4x - 6$, find the resulting expression from performing the operation $2 \cdot A(x) - B(x)$.
The height in feet of a drone as a function of time in seconds is given by the polynomial $H(t) = -5t^2 + 20t + 2$. A second drone follows a path described by…
The height in feet of a drone as a function of time in seconds is given by the polynomial $H(t) = -5t^2 + 20t + 2$. A second drone follows a path described by $G(t) = -5t^2 + 15t + 10$.
Find the polynomial $D(t)$ that represents the difference in height between the first and second drone ($D(t) = H(t) - G(t)$). At what time $t$ are both drones at the same height?
Multiplying polynomials
To multiply polynomials, use the distributive property: every term of the first polynomial multiplies every term of the second. Then combine like terms. When you multiply powers of $x$, the exponents add: $x \cdot x^2 = x^3$.
An area model shows why: each product is the area of one small rectangle.
For a product of three polynomials, multiply two of them first and then multiply that result by the third.
Multiply the polynomials $P(x) = x^2 - 3x + 2$ and $Q(x) = 2x - 5$. Express the result ordered from highest to lowest degree.
Multiply the polynomials $P(x) = x^2 - 3x + 2$ and $Q(x) = 2x - 5$. Express the result ordered from highest to lowest degree.
A cardboard box without a lid is to be made from a rectangular sheet $30 \text{ cm}$ long by $20 \text{ cm}$ wide. To do this, four identical squares of side…
A cardboard box without a lid is to be made from a rectangular sheet $30 \text{ cm}$ long by $20 \text{ cm}$ wide. To do this, four identical squares of side $x \text{ cm}$ are cut out from each of the corners and the flaps are folded up. Find the polynomial $V(x)$ that expresses the volume of the resulting box in terms of $x$.
Squaring a binomial
Squaring a binomial shows up so often that it pays to recognize the pattern. Multiply $(a + b)(a + b)$ and the two middle terms $ab$ and $ba$ add up to $2ab$.
The square of side $a + b$ splits into a big square, a small square and two equal rectangles. Move the sliders and watch.
Cubing a binomial
Multiply $(a \pm b)^2$ by one more $(a \pm b)$ and you get:
For example, $(x - 1)^3 = x^3 - 3x^2(1) + 3x(1)^2 - 1 = x^3 - 3x^2 + 3x - 1$.
Dividing by a monomial
To divide a polynomial by a monomial, divide each term by it. For example,
$$\frac{12x^2y - 8xy^2}{-4y} = \frac{12x^2y}{-4y} + \frac{-8xy^2}{-4y} = -3x^2 + 2xy.$$Expand the following expressions by applying the corresponding algebraic identities: a) $(3x + 4)^2$ b) $(2x - 5)^2$
Expand the following expressions by applying the corresponding algebraic identities:
- a) $(3x + 4)^2$
- b) $(2x - 5)^2$
Expand and fully simplify the following algebraic expression, paying special attention to signs and algebraic identities: $3(x - 2)^2 - 2(x + 1)(x - 1) + (x + 3)^2$
Expand and fully simplify the following algebraic expression, paying special attention to signs and algebraic identities:
$3(x - 2)^2 - 2(x + 1)(x - 1) + (x + 3)^2$
Simplify the following algebraic expression, factoring the final result if possible: $\frac{(2x - y)^3 - (2x + y)^3}{-2y}$ Then, evaluate the resulting polynomial for…
Simplify the following algebraic expression, factoring the final result if possible:
$\frac{(2x - y)^3 - (2x + y)^3}{-2y}$
Then, evaluate the resulting polynomial for $x = 2$ and $y = -1$.
Difference of squares
When you multiply a sum by the matching difference, the middle terms cancel:
$(a + b)(a - b) = a^2 - ab + ab - b^2 = a^2 - b^2$.
Proving an identity
To prove that two expressions are equal for every $x$, simplify each side and show they match. For instance, $(x + 1)^2 - (x - 1)^2 = x^2 + 2x + 1 - x^2 + 2x - 1 = 4x$.
The same idea shows that three lengths $a$, $b$, $c$ form a right triangle: by the converse of the Pythagorean Theorem, you only need to show that $a^2 + b^2 = c^2$. Simplify the left side, simplify the right side, and compare. You can always check your algebra with a number.
Express the following binomials as a product of two factors using the difference of two squares identity: a) $16x^2 - 25$ b) $9x^4 - 49$
Express the following binomials as a product of two factors using the difference of two squares identity:
- a) $16x^2 - 25$
- b) $9x^4 - 49$
Expand and fully simplify the following algebraic expression: $(x + 3)(x - 3) - (x - 2)^2$.
Expand and fully simplify the following algebraic expression: $(x + 3)(x - 3) - (x - 2)^2$.
Prove algebraically that, for any real number $x > 1$, the expressions $a = x^2 - 1$, $b = 2x$, and $c = x^2 + 1$ always form the sides of a right triangle (that is,…
Prove algebraically that, for any real number $x > 1$, the expressions $a = x^2 - 1$, $b = 2x$, and $c = x^2 + 1$ always form the sides of a right triangle (that is, they satisfy the Pythagorean theorem). Check your result by evaluating the polynomials for $x = 4$.
Polynomial long division
Dividing polynomials works like dividing whole numbers. Since $7 \div 2$ gives quotient $3$ and remainder $1$, we write $7 = 2 \cdot 3 + 1$. For polynomials:
$$D(x) = d(x) \cdot C(x) + R(x)$$where $D$ is the dividend, $d$ the divisor, $C$ the quotient and $R$ the remainder. The remainder must have a smaller degree than the divisor (or be $0$).
If the dividend skips a power, write it with a coefficient of $0$ as a placeholder, for example $x^3 - 7x + 6 = x^3 + 0x^2 - 7x + 6$. When the remainder is $0$, the division is exact: the divisor is a factor and $D(x) = d(x) \cdot C(x)$.
Divide the polynomial $D(x) = 4x^4 - 2x^3 + 6x^2 - x + 5$ by the binomial $d(x) = x^2 + 2$. Clearly indicate the quotient polynomial $C(x)$ and the remainder polynomial…
Divide the polynomial $D(x) = 4x^4 - 2x^3 + 6x^2 - x + 5$ by the binomial $d(x) = x^2 + 2$. Clearly indicate the quotient polynomial $C(x)$ and the remainder polynomial $R(x)$.
Determine if the polynomial $P(x) = x^4 - 5x^3 + 7x^2 - 5x + 6$ is divisible by the binomial $(x^2 + 1)$ by performing the division. If so, write the factorization of…
Determine if the polynomial $P(x) = x^4 - 5x^3 + 7x^2 - 5x + 6$ is divisible by the binomial $(x^2 + 1)$ by performing the division. If so, write the factorization of $P(x)$ as a product of two polynomials.
Determine the values of the coefficients $m$ and $n$ so that the division of the polynomial $D(x) = 4x^4 - 2x^3 + mx^2 + nx - 5$ by the divisor $d(x) = 2x^2 - x + 3$ is…
Determine the values of the coefficients $m$ and $n$ so that the division of the polynomial $D(x) = 4x^4 - 2x^3 + mx^2 + nx - 5$ by the divisor $d(x) = 2x^2 - x + 3$ is exact.
Synthetic division
When the divisor has the form $x - a$, there is a much faster shortcut called synthetic division. You work only with the coefficients. (Some problems call this Ruffini's rule; it is exactly the same method.)
You can also go backwards. If you know the divisor, the quotient and the remainder, rebuild the dividend with $P(x) = (x - a) \cdot C(x) + R$.
Apply Ruffini's rule to divide the polynomial $P(x) = 3x^3 - 5x^2 + 2x - 7$ by $(x - 2)$. Write the quotient and the remainder obtained.
Apply Ruffini's rule to divide the polynomial $P(x) = 3x^3 - 5x^2 + 2x - 7$ by $(x - 2)$. Write the quotient and the remainder obtained.
Given that dividing a polynomial $P(x)$ by $(x - 3)$ gives a quotient of $C(x) = 4x^2 - 2x + 1$ and a remainder of $R = -7$, find the polynomial $P(x)$ and express it in…
Given that dividing a polynomial $P(x)$ by $(x - 3)$ gives a quotient of $C(x) = 4x^2 - 2x + 1$ and a remainder of $R = -7$, find the polynomial $P(x)$ and express it in expanded form.
The Remainder Theorem
Do you always need to divide to find the remainder? No. Start from $P(x) = (x - a) \cdot Q(x) + R$ and plug in $x = a$: $P(a) = 0 \cdot Q(a) + R = R$.
With two unknown coefficients, each condition gives one equation, and you solve the system.
Find the value of the parameter $k$ so that dividing the polynomial $P(x) = 2x^3 - kx^2 + 5x - 4$ by the binomial $(x + 2)$ gives a remainder equal to $10$. Use the…
Find the value of the parameter $k$ so that dividing the polynomial $P(x) = 2x^3 - kx^2 + 5x - 4$ by the binomial $(x + 2)$ gives a remainder equal to $10$. Use the remainder theorem to justify your answer.
Find the values of the parameters $a$ and $b$ so that the polynomial $P(x) = x^3 + ax^2 + bx - 6$ is exactly divisible by $(x - 2)$ and, at the same time, leaves a…
Find the values of the parameters $a$ and $b$ so that the polynomial $P(x) = x^3 + ax^2 + bx - 6$ is exactly divisible by $(x - 2)$ and, at the same time, leaves a remainder of $-12$ when divided by $(x + 1)$.
The Factor Theorem and zeros
A number $a$ with $P(a) = 0$ is a zero (or root) of $P$. By the Remainder Theorem, the remainder when dividing by $x - a$ is then $0$. That gives us the Factor Theorem.
The polynomial from the first section, $x^3 - 4x^2 + x + 6$, had $P(-1) = 0$ and $P(2) = 0$. Its graph crosses the $x$-axis at those values (and at $3$).
Factoring when you know one zero: divide by $x - a$ with synthetic division, then factor the quotient.
Building a polynomial from its zeros: each zero gives a factor, but you also need a number that fixes the leading coefficient.
A polynomial $P(x)$ of degree 3 has roots $x = 1$, $x = -2$, and $x = 4$. Given that the value of the polynomial for $x = 0$ is equal to $16$, find the expanded form of…
A polynomial $P(x)$ of degree 3 has roots $x = 1$, $x = -2$, and $x = 4$. Given that the value of the polynomial for $x = 0$ is equal to $16$, find the expanded form of $P(x)$ (that is, in the form $ax^3 + bx^2 + cx + d$).
Geometry with polynomials
Lengths like $x + 3$ or $2x - 1$ turn areas and volumes into polynomials. Write the formula with the given lengths, multiply, and combine like terms.
A rectangular garden is $(2x + 5)$ feet long and $(2x - 5)$ feet wide. Inside it, a square fountain with side length $(x - 1)$ feet will be built, and the rest will be…
A rectangular garden is $(2x + 5)$ feet long and $(2x - 5)$ feet wide. Inside it, a square fountain with side length $(x - 1)$ feet will be built, and the rest will be covered with grass. Express the area of the lawn area as a simplified polynomial in $x$.
A running track consists of a central rectangular region of length $3x$ and width $2x$, with two semicircles at its ends of diameter $2x$. Express the total area and the…
A running track consists of a central rectangular region of length $3x$ and width $2x$, with two semicircles at its ends of diameter $2x$. Express the total area and the outer perimeter of the track as polynomials in terms of $x$ (leave $\pi$ in the coefficients).
A water tank is cylindrical with a base radius of $(x + 2)$ feet and a height of $(2x - 1)$ feet. A solid cubic block with edge length $x$ feet is submerged inside it.…
A water tank is cylindrical with a base radius of $(x + 2)$ feet and a height of $(2x - 1)$ feet. A solid cubic block with edge length $x$ feet is submerged inside it. Express, using a polynomial in standard form, the volume of water needed to fill the remaining space in the tank. Leave the constant $\pi$ in the coefficients that require it.
Profit and other applications
In business, revenue is quantity times price, and profit is revenue minus cost. Each is a polynomial, so the profit is one too.
Now let's close the hook from the start of the page. The robotics club has cost $C(x) = 15x + 300$ and price $p(x) = 50 - 0.5x$ per hoodie.
Back to the start: the club breaks even at $10$ hoodies. It makes a profit between $10$ and $60$ hoodies, and at $60$ hoodies the low price eats all the profit again.
The production cost of $x$ units of an electronic device is given by the polynomial $C(x) = 50x + 200$. The retail price of each unit depends on the quantity…
The production cost of $x$ units of an electronic device is given by the polynomial $C(x) = 50x + 200$. The retail price of each unit depends on the quantity manufactured and follows the expression $p(x) = 300 - 2x$.
Find the profit polynomial $B(x)$, knowing that the profit is equal to total revenue (quantity times price) minus total costs. What will the profit be if $40$ units are manufactured and sold?
A startup estimates that its monthly revenue follows the polynomial $I(x) = -2x^3 + 60x^2$ and its operating costs are $C(x) = 15x^2 + 400x$, where $x$ is the number of…
A startup estimates that its monthly revenue follows the polynomial $I(x) = -2x^3 + 60x^2$ and its operating costs are $C(x) = 15x^2 + 400x$, where $x$ is the number of months elapsed since its creation.
Find the profit polynomial $B(x)$. Given that in month $x = 10$ the profit is exactly zero (break-even point), factor the polynomial $B(x)$ to determine in which other month the profit is zero again.