Polynomials
A water rocket launches from a platform. After $t$ seconds, its height in feet is $h(t) = -5t^2 + 25t + 2$. One formula tells you the height at any moment. That formula is a polynomial. In this page you will learn to read, evaluate, combine, multiply and factor polynomials.
What is a polynomial?
You already know monomials: a number, a variable, or a product of them, like $9x^3$ or $-\tfrac{1}{2}x^2y$. Each monomial has three parts.
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Coefficient: the number in front. For $m$ it is $1$, and for $-x$ it is $-1$.
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Variable part: the letters with their exponents.
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Degree: the sum of the exponents of the variables. $-\tfrac{1}{2}x^2y$ has degree $2 + 1 = 3$. A plain number like $-4$ has degree $0$.
| Monomial | Coefficient | Variable part | Degree |
|---|---|---|---|
| $9x^3$ | $9$ | $x^3$ | $3$ |
| $-\tfrac{1}{2}x^2y$ | $-\tfrac{1}{2}$ | $x^2y$ | $3$ |
| $m$ | $1$ | $m$ | $1$ |
| $-4$ | $-4$ | none | $0$ |
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Degree of the polynomial: the highest degree of any term (here, $4$).
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Leading coefficient: the coefficient of that highest-degree term ($-5$).
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Constant term: the term with no variable ($9$).
A subtraction is a sum with a negative term. In $x^2 - 3x$ the terms are $x^2$ and $-3x$.
Standard form
A polynomial is usually written in standard form: terms from highest degree to lowest. It is complete if every degree from the highest down to the constant appears. Otherwise it is incomplete. You can fill a missing degree with a $0$ term. For example, $x^3 - 4$ is incomplete, and $x^3 + 0x^2 + 0x - 4$ is the same polynomial with the gaps filled.
Copy and complete the following table in your notebook to identify the fundamental elements of these monomials: | Monomial | Coefficient | Variable Part | Degree | |…
Copy and complete the following table in your notebook to identify the fundamental elements of these monomials:
| Monomial | Coefficient | Variable Part | Degree |
|---|---|---|---|
| $-7x^4$ | |||
| $\frac{2}{3}ab^2$ | |||
| $xyz$ | |||
| $5$ |
Analyze the following polynomial and answer the questions: $P(x) = -2x^3 + 5x^2 - 8$ a) What is the degree of the polynomial? b) What is the name of the term that has no…
Analyze the following polynomial and answer the questions:
$P(x) = -2x^3 + 5x^2 - 8$
- a) What is the degree of the polynomial?
- b) What is the name of the term that has no variable (the $-8$)?
- c) How many terms does it have? According to the number of terms, is it a monomial, binomial, or trinomial?
- d) What is the coefficient of the term with the highest degree?
Evaluating a polynomial and its roots
Dog walking pays $\$6$ to start plus $\$4$ for each dog, so the pay for $x$ dogs is $E = 6 + 4x$ dollars. For $5$ dogs, substitute $x = 5$: $E = 6 + 4(5) = 26$ dollars. That is the whole idea of evaluating.
Why parentheses? Because the number is replaced as a whole. Squaring $-3$ means $(-3)^2 = 9$, not $-3^2 = -9$.
Now back to the rocket. Its height at launch is $h(0) = -5(0)^2 + 25(0) + 2 = 2$ feet, so the platform is 2 feet above the ground. One second later, $h(1) = -5 + 25 + 2 = 22$ feet. Explore the graph below (here $x$ plays the role of time $t$).
Roots
A root can also help you find a missing coefficient.
A pizza delivery driver earns a fixed wage of $\$15$ for each afternoon of work, plus an extra $\$3$ for each home delivery order. The expression that calculates their…
A pizza delivery driver earns a fixed wage of $\$15$ for each afternoon of work, plus an extra $\$3$ for each home delivery order. The expression that calculates their earnings is $G = 15 + 3x$, where $x$ is the number of orders.
Calculate the numerical value of their earnings (what they will earn in total) if one afternoon they manage to deliver $8$ orders.
Given the following polynomial: $P(x) = -5x^4 + \frac{2}{3}x^2 - x + 9$ State with reasoning the following elements: a) The degree of the polynomial. b) The coefficient…
Given the following polynomial:
$P(x) = -5x^4 + \frac{2}{3}x^2 - x + 9$
State with reasoning the following elements:
- a) The degree of the polynomial.
- b) The coefficient of the degree 2 term.
- c) The constant term.
- d) The numerical value of $P(x)$ for $x = -1$.
A small water rocket is launched from a platform. The height $h$ (in feet) of the rocket as a function of time $t$ (in seconds) is given by the polynomial:…
A small water rocket is launched from a platform. The height $h$ (in feet) of the rocket as a function of time $t$ (in seconds) is given by the polynomial:
$h(t) = -5t^2 + 25t + 2$
- a) At what height is the launch platform? (Hint: calculate $h(0)$).
- b) At what height will the rocket be 2 seconds after being launched?
- c) Will the rocket be higher at 3 seconds or at 4 seconds? Justify your answer by calculating the corresponding numerical values.
Given the polynomial $P(x) = x^3 - kx^2 + 5x - 6$, determine the value of the coefficient $k$ knowing that the numerical value of the polynomial for $x = 2$ is equal to…
Given the polynomial $P(x) = x^3 - kx^2 + 5x - 6$, determine the value of the coefficient $k$ knowing that the numerical value of the polynomial for $x = 2$ is equal to zero ($x = 2$ is a root of the polynomial). Once $k$ is found, calculate the numerical value of $P(x)$ for $x = -1$.
Like terms
You can add $3$ apples and $2$ apples, but not $3$ apples and $2$ oranges. Terms work the same way.
To combine like terms, add their coefficients and keep the variable part: $3x^2 - 5x^2 = -2x^2$.
To be able to add or subtract monomials, they must be like terms. Select from the following list which ones are like terms to the monomial $5x^2y$: $5xy^2$ $-x^2y$…
To be able to add or subtract monomials, they must be like terms. Select from the following list which ones are like terms to the monomial $5x^2y$:
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$5xy^2$
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$-x^2y$
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$2x^2$
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$\frac{1}{2}x^2y$
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$5x^2y^2$
Adding and subtracting polynomials
To add polynomials, drop the parentheses, then combine like terms. To subtract, add the opposite: change the sign of every term in the polynomial being subtracted. You can also stack the polynomials in columns, lining up like terms.
To multiply a polynomial by a number, distribute: $2(x^3 - 3x + 1) = 2x^3 - 6x + 2$.
Polynomials show up in perimeters, because perimeter is the sum of all the sides.
For this rectangle, $P = (2x + 5) + (x + 3) + (2x + 5) + (x + 3) = 6x + 16$. Equivalent shortcut: $P = 2(2x + 5) + 2(x + 3) = 4x + 10 + 2x + 6 = 6x + 16$.
Given the following polynomials, work out the sum $A(x) + B(x)$ and combine like terms: $A(x) = 4x^2 - 5x + 3$ $B(x) = x^2 + 2x - 7$
Given the following polynomials, work out the sum $A(x) + B(x)$ and combine like terms:
$A(x) = 4x^2 - 5x + 3$
$B(x) = x^2 + 2x - 7$
Perform the following operations with polynomials and simplify the final result: a) $(4x^3 - 5x^2 + x - 7) + (x^3 + 2x^2 - 10)$ b) $(3x^2 - 6x + 2) - (x^2 - 4x - 5)$ c)…
Perform the following operations with polynomials and simplify the final result:
- a) $(4x^3 - 5x^2 + x - 7) + (x^3 + 2x^2 - 10)$
- b) $(3x^2 - 6x + 2) - (x^2 - 4x - 5)$
- c) $2 \cdot (x^3 - 3x + 1) + 4 \cdot (x^2 - 2)$
Find the simplified algebraic expression that represents the perimeter of the following rectangular photo frame:
Find the simplified algebraic expression that represents the perimeter of the following rectangular photo frame:
Writing polynomials from situations
Real problems give you words. Your job is to turn them into an expression, simplify it, and then evaluate it when you know $x$.
Coefficients can be letters too. A ride-share fare might be a booking fee $a$, plus $b$ dollars per mile, plus a long-trip surcharge of $\tfrac{x^2}{10}$ for $x$ miles: $F(x) = a + bx + \tfrac{1}{10}x^2$. Slide $a$ and $b$ to see how each changes the fare.
If the price per mile doubles, only the term with $b$ changes: $F_2(x) = a + 2bx + \tfrac{1}{10}x^2$.
In the world of video games, the scoring system of a new platform title works as follows: for each gold coin collected you receive an unknown amount of points $x$, but…
In the world of video games, the scoring system of a new platform title works as follows: for each gold coin collected you receive an unknown amount of points $x$, but for each enemy defeated you receive three times as many points as for a coin. At the end of the level, you are always given a fixed bonus of $50$ experience points.
Write the algebraic expression that represents the total score of a player who has collected $12$ coins and defeated $4$ enemies.
In an electronics store, the price of a mid-range smartphone is $\$x$. A high-end smartphone costs twice as much as the mid-range one plus $\$150$, and a tablet costs…
In an electronics store, the price of a mid-range smartphone is $\$x$. A high-end smartphone costs twice as much as the mid-range one plus $\$150$, and a tablet costs $\$80$ less than the mid-range smartphone.
- a) Write the algebraic expression that represents the total cost of buying two mid-range smartphones, one high-end smartphone, and three tablets.
- b) Simplify the obtained expression so that it is a polynomial in its simplest form.
- c) If the price of the basic smartphone is $\$x = 200$, how much money will the total order cost?
A courier company charges for sending parcels according to the following rate: a fixed handling cost of $\$G$, plus a price for each pound of weight $x…
A courier company charges for sending parcels according to the following rate: a fixed handling cost of $\$G$, plus a price for each pound of weight $x$, and a "premium distance" surcharge which is equivalent to the square of the parcel's weight divided by 4.
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Write the polynomial $C(x)$ that represents the total cost of sending a parcel of $x$ pounds.
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If the handling fees are $\$G = 5$ and we want to send a parcel of $8$ pounds, what will the total cost be?
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If the company decides to double the cost per pound and keep the rest of the parameters, write the new polynomial $C_2(x)$.
Multiplying polynomials
To multiply polynomials, use the distributive property: multiply each term of one polynomial by each term of the other, then combine like terms. When you multiply powers of the same variable, add the exponents: $x \cdot x^2 = x^3$.
An area model shows why. A rectangle with sides $x + 3$ and $x + 2$ splits into four pieces.
Some products simplify a lot. Try $(x - 1)(x^2 + x + 1)$.
The same thing happens with $(x + 2)(x^2 - 2x + 4) = x^3 + 8$. When you see a binomial times a trinomial like this, expect the middle terms to cancel.
Multiply the following polynomials using the distributive property and combine like terms: a) $P(x) = (2x - 3) \cdot (x^2 + 4x - 5)$ b)…
Multiply the following polynomials using the distributive property and combine like terms:
- a) $P(x) = (2x - 3) \cdot (x^2 + 4x - 5)$
- b) $Q(x) = (x^2 + 1) \cdot (3x^2 - 2x + 4)$
Special products
Some products come up so often that you should recognize their pattern. Look at $(a + b)^2 = (a + b)(a + b) = a^2 + ab + ba + b^2$. The two middle terms are like terms, so they combine into $2ab$.
In the third pattern the middle terms $-ab$ and $+ab$ cancel. Some problems call these patterns algebraic identities; they are just shortcuts for multiplying.
A special product can also make a longer problem shorter. To expand $(x + 3)^2(x - 3)$, first write the square as two factors and notice $(x + 3)(x - 3) = x^2 - 9$. Then the whole expression is $(x + 3)(x^2 - 9) = x^3 + 3x^2 - 9x - 27$.
Combined problems use both tools: special products for the squares, and the distributive property for the rest.
Algebraic identities allow us to quickly calculate some powers. Use the formula for the square of a sum $(a + b)^2 = a^2 + 2ab + b^2$ to expand the following expression:…
Algebraic identities allow us to quickly calculate some powers. Use the formula for the square of a sum $(a + b)^2 = a^2 + 2ab + b^2$ to expand the following expression:
$(x + 2)^2$
Apply the formulas for algebraic identities to expand the following expressions: a) $(3x + 4)^2$ b) $(2x^2 - 5)^2$ c) $(x + 7) \cdot (x - 7)$ d)…
Apply the formulas for algebraic identities to expand the following expressions:
- a) $(3x + 4)^2$
- b) $(2x^2 - 5)^2$
- c) $(x + 7) \cdot (x - 7)$
- d) $(\frac{1}{2}x + 6) \cdot (\frac{1}{2}x - 6)$
Work out the following combined operation with polynomials and combine like terms: $$ [ (x + 2)^2 \cdot (x - 2) ] - (x - 1) \cdot (x^2 + x + 1) $$ Hint: Look closely to…
Work out the following combined operation with polynomials and combine like terms:
$$ [ (x + 2)^2 \cdot (x - 2) ] - (x - 1) \cdot (x^2 + x + 1) $$Hint: Look closely to see if you can apply any algebraic identity before multiplying.
Area problems
Rectangle area is length $\times$ width, so polynomials multiply here. For a shape with a hole, subtract the area of the hole from the area of the whole. Put the hole's area in parentheses before you subtract.
The same idea works for growth. Subtract the old area from the new area to find the increase.
Calculate the area of the shaded region in the following figure. Express the result as a simplified polynomial in terms of $x$.
Calculate the area of the shaded region in the following figure. Express the result as a simplified polynomial in terms of $x$.
A park has the shape of a rectangle. Originally, its base measures $x$ feet and its height measures $5$ feet less than the base. The city council decides to expand the…
A park has the shape of a rectangle. Originally, its base measures $x$ feet and its height measures $5$ feet less than the base. The city council decides to expand the park by increasing the base by $3$ feet and the height by $2$ feet.
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Write the polynomial $A_1(x)$ for the area of the original park.
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Write the polynomial $A_2(x)$ for the area of the expanded park.
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Find the polynomial that represents the increase in area $I(x) = A_2(x) - A_1(x)$ and simplify it.
Calculate the area of the shaded region in the following figure, expressing the result as a simplified polynomial in terms of $x$. The outer figure is a square of side…
Calculate the area of the shaded region in the following figure, expressing the result as a simplified polynomial in terms of $x$. The outer figure is a square of side $3x + 2$ and the inner hole is a rectangle of base $x + 4$ and height $x - 1$.
Patterns that grow
A video game designer builds a floor pattern in levels. To write a polynomial for a pattern, count the tiles for levels $1$, $2$, $3$. Look for a rule, check it against the picture, and write it with $n$.
Here is a pool of $n \times n$ tiles with a border of one tile around it.
A video game designer is creating a tile pattern for a dungeon floor. In level 1 they use a central tile, and in each subsequent level they surround the previous pattern…
A video game designer is creating a tile pattern for a dungeon floor. In level 1 they use a central tile, and in each subsequent level they surround the previous pattern with a new layer of tiles of a different color. The total number of tiles in level $n$ follows a logical sequence.
Look at the figure representing the first three levels and:
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Find the polynomial $P(n)$ that expresses the total number of tiles needed to build up to level $n$.
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Calculate how many tiles a level $n = 12$ floor will have.
Factoring
Factoring is multiplying in reverse. You rewrite a polynomial as a product.
Step 1: Factor out the GCF
Find the greatest common factor (GCF) of all the terms. That means the largest number that divides every coefficient, times the lowest power of each variable they all share. Write it outside the parentheses and put what is left inside.
With two letters, do the same for each: $8a^3b^2 - 20a^2b = 4a^2b(2ab - 5)$. Fractions work too: $\tfrac{1}{4}x^3 + \tfrac{3}{4}x = \tfrac{1}{4}x(x^2 + 3)$.
Step 2: Look for a special product
Read the special products backward.
A trinomial is a perfect square when the first and last terms are squares and the middle term is twice the product of their roots. For example, $x^2 + 14x + 49$: $x^2 = x^2$, $49 = 7^2$, and $2 \cdot x \cdot 7 = 14x$ ✓. So it equals $(x + 7)^2$.
Simplifying with these tools
Special products and factoring together simplify messy expressions.
Some problems say "factor out the common factor." That is exactly the GCF step above.
Factoring out a common factor is the reverse step of the distributive property. Use this technique to simplify the following algebraic expressions: a)…
Factoring out a common factor is the reverse step of the distributive property. Use this technique to simplify the following algebraic expressions:
- a) $12x^5 - 8x^3 + 4x^2$
- b) $5a^2b - 15ab^2 + 10ab$
- c) $\frac{1}{3}x^4 - \frac{2}{3}x^2$
Factor out the common factor and use algebraic identities to factor the following polynomials: 1. $Q(x) = 3x^3 - 12x$ 2. $R(x) = x^4 - 10x^3 + 25x^2$ 3.…
Factor out the common factor and use algebraic identities to factor the following polynomials:
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$Q(x) = 3x^3 - 12x$
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$R(x) = x^4 - 10x^3 + 25x^2$
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$S(x) = 2x^2 + 20x + 50$
Work out and fully simplify the following algebraic expression using algebraic identities and factoring out the common factor: $$ \frac{(2x + 3)^2 - (2x - 3)^2}{12x} $$…
Work out and fully simplify the following algebraic expression using algebraic identities and factoring out the common factor:
$$ \frac{(2x + 3)^2 - (2x - 3)^2}{12x} $$State whether the final result is a constant number or depends on the value of $x$.
Summary
Back to the start: the rocket's $h(t) = -5t^2 + 25t + 2$ is a trinomial of degree $2$. Its constant term, $2$, is the platform height, and its value at any time gives the rocket's height at that moment.