1.2 Polynomials

Terri Manthey

1.2 Polynomials

The word polynomial comes from poly- meaning many and -nomial from binomial which means having two names. Binomial was actually defined first as 2 algebraic expressions separated by a + or – sign. Then, polynomial means any number of algebraic expressions separated by + or – signs. These are not just any algebraic expressions. The algebraic expressions must follow a very important rule to be considered polynomials. The rule is that all the exponents on the variables have to be non-negative integers. This means that no variable can be in the denominator nor can any variable by under a root sign.

A number multiplied to a variable raised to an exponent is known as the coefficient. These coefficients can be positive, negative, zero, and they can be whole numbers, fractions, square roots or other irrational numbers like $\pi$ . We should be able to write each product of the variable and coefficient part as $a_i x^i$ , and are called a term of a polynomial. Some examples of terms are $3x^2, \;x,\; y\sqrt{2}, \;8 \pi t^2,\; \frac{1}{12}w^{10}, \;221$ . Notice the last example does not seem to have a variable part, remember that $x^0=1$ so terms with a number only part are allowed and are called constants or constant terms.

A polynomial containing only one term, such as $9xy^2$ is called a monomial. As the name suggests, if a polynomial has two terms, it is called a binomial, and if it contains three terms, it is called a trinomial.

The degree of a polynomial is the highest power of the variable that occurs in any term. If there are more than one variable, then we add the exponents for the term and the highest of those sums is the degree. The term with the highest degree is called the leading term and its coefficient is called the leading coefficient. The standard form of a polynomial is with the leading term first and then descending powers of one variable. Consider a polynomial in one variable, let’s say $x$ , it might look like this:

A polynomial reading: a sub n times x to the nth power plus and so on plus a sub 2 times x squared plus a sub one times x plus a subzero is shown. The a in the term a sub n is labeled: leading coefficient. The n in the term x to the nth power is labeled: degree. Finally, the entire term is labeled as: Leading term.

Polynomial in one variable

A polynomial is an expression that can be written in the form

$a_nx^n+ . . . +a_2x^2+a_1x+a_0.$

Each real number $a_i$ is called a coefficient. The number $a_0$ that is not multiplied by a variable is called a constant. Each product $a_ix^i$ is a term of a polynomial. The highest power of the variable that occurs in the polynomial is called the degree of a polynomial. The leading term is the term with the highest power, and its coefficient is called the leading coefficient.

Given a polynomial expression, we can easily identify the degree and leading coefficient by:

Rewrite the expression in descending powers of the variable, being careful to keep the + or – in front of them.
The exponent on the variable in the first term will be your degree.
The coefficient of the first term will be your lead coefficient.

Examples on finding the degree and the leading coefficient of a polynomial

a. $3+2x^2-4x^3$

Rewrite in descending powers of $x$ , notice we keep the negative sign on the $-4x^3$ and we have to put a + on the 3.

$-4x^3+2x^2+3$

Now we can see that the exponent on the first term is 3, so the degree is 3 and the lead coefficient is -4.

b. $5t^5-2t^3+7t$

Notice that this one is already in descending powers of $t$ , so we can see that the degree is 5 and the lead coefficient is also 5.

c. $6p-p^3-2$

We need to rewrite this in descending powers of p: $-p^3+6p-2$ , now we can see that the degree is 3 and the lead coefficient is understood as -1.

Try It Now 1

Identify the degree, leading term, and leading coefficient of the polynomial

$4x^2-x^6+2x-6$

Adding and Subtracting Polynomials

We can add and subtract polynomials by combining like terms, which are terms that contain the same variables raised to the same exponents. For example, $5x^2$ and $-2x^2$ are like terms, and can be added to get $3x^2$ , but $3x$ and $3x^2$ are not like terms and therefore cannot be added.

Warning Triangle with Exclamation Point Warning! Be extra careful when unlike terms look tricky. 2-5x cannot be combined. 2 is a constant and 5 has to multiplied to x first.

Adding Polynomials

Given two or more polynomials that we need to add, we can simply drop the parentheses and add the like terms. Do not forget to simplify and write them back in standard (descending powers of the variable) form.

Examples of adding polynomials

$(12x^2+9x-21)+(4x^3+8x^2-5x+20)$

Drop the ( )’s: $12x^2+9x-21+4x^3+8x^2-5x+20$
Combine like terms. One easy way to do this is to write them vertically:

$\begin{array}{rrrr} 4x^3 & +8x^2 & -5x & +20 \\ \; & +12x^2 & +9x & -21 \\ \hline4x^3 & +20x^2 & +4x & -1 \end {array}$

We can check our answers to these types of problems using a graphing calculator or Desmos. Just put the problem into one line and your answer on the other. If they are the same graph, then your answer is correct. Most technology will not do the algebra for us, but we can make sure we are correct.

Try it Now 2

$(2{x^3} + 5{x^2} - x + 1) + (2{x^2} - 3x - 4)$

Subtracting Polynomials

Given two or more polynomials that we need to subtract, we must distribute the negative through the second polynomial before we can drop the parentheses and add the like terms. Distributing the negative means we make everything in the second polynomial the opposite sign. Do not forget to simplify and write them back in standard (descending powers of the variable) form.

Example of Subtracting Polynomials

$(7x^4-x^2+6x+1)-(5x^3-2x^2+3x+2)$

Make the second polynomial opposites and drop the ( )’s: $7x^4-x^2+6x+1-5x^3+2x^2-3x-2$
Combine like terms. One easy way to do this is to write them vertically:

$\begin{array}{rrrrr} 7x^4 & \; & -x^2 & +6x & +1 \\ \; & -5x^3 & +2x^2 & -3x & -2 \\ \hline{7x^4 & -5x^3 & +x^2 & +3x & -1 \end {array}$

Try it Now 3

$(-7x^3-7x^2+6x-2)-(4x^3-6x^2-x+7)$

Multiplying Polynomials

Multiplying polynomials is a bit more challenging than adding and subtracting polynomials. We must use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. We then combine like terms. We can also use a shortcut called the FOIL method when multiplying binomials. Certain special products follow patterns that we can memorize and use instead of multiplying the polynomials by hand each time. We will look at a variety of ways to multiply polynomials.

Multiplying Polynomials Using the Distributive Property

To multiply a number by a polynomial, we use the distributive property. The number must be distributed to each term of the polynomial. We can distribute the $2$ in $2(x+7)$ to obtain the equivalent expression $2x+14$ . When multiplying polynomials, the distributive property allows us to multiply each term of the first polynomial by each term of the second. We then add the products together and combine like terms to simplify.

We will look at three methods to multiply two polynomials, but it all boils down to multiplying each term of the first polynomial by each term of the second and then combine the like terms.

Method 1: FOIL

This only works when you are multiplying two binomials. First-Outer-Inner-Last

Two quantities in parentheses are being multiplied, the first being: a times x plus b and the second being: c times x plus d. This expression equals ac times x squared plus ad times x plus bc times x plus bd. The terms ax and cx are labeled: First Terms. The terms ax and d are labeled: Outer Terms. The terms b and cx are labeled: Inner Terms. The terms b and d are labeled: Last Terms.

Many times, we take this one step further if $a$ , $b$ , $c$ , and $d$ are all real numbers and combine our two x terms: $acx^2+(ad+bc)x+bd$ .

Method 2: Lattice

The lattice method works by creating a little table. Let’s say we are multiplying $3x^2-4x+5$ and $x+2$ . We write one polynomial across the top and the other down the left side.

So combining $-4x^2+6x^2 = 2x^2$ and $5x-8x=-3x$ while $3x^3$ and $10$ do not combine to anything. So, written in descending powers of $x$ , we end up with $3x^3+2x^2-3x+10$ .

Method 3: Old-fashioned Multiplication

Start by writing the two polynomials vertically similar to how you might multiply two larger numbers.

Remember multiplying two 2 or 3 digit numbers and note how we can similarly multiply two polynomials.

Examples Multiplying Polynomials

a. $(x+3)(2x-5)$

Let’s use Foil:

First: $x \cdot 2x=2x^2$
Outer: $x \cdot -5=-5x$
Inner: $3 \cdot 2x=6x$
Last: $3 \cdot -5=-15$

Now we have $2x^2-5x+6x-15$ and combining like terms in the middle, we have $2x^2+x-15$ .

b. $(x^2-5x+6)(3x-4)$

Let’s create our table:

	$3x$	$-4$
$x^2$	$3x^3$	$-4x^2$
$-5x$	$-15x^2$	$20x$
$6$	$18x$	$-24$

Now we just combine the like terms on the diagonals and we get $3x^3-19x^2+38x-24$ .

c. $(5y-18)(3y-2)$

Let’s try the vertical method with this one:

$\begin{array}{rrrll} 5y -18 \\ 3y - 2 \\ \hline{-10y + 36} \\ 15y^2-54y \qquad\\ \hline{15y^2-64y+36} \end {array}$

Try it Now 4

a. $(3t-8)(t-9)$

b. $\left(2x^2+7x-4 \right) \left( 5x-2 \right)$

c. $(3y-21)(5y+6)$

Perfect Square Trinomials

Certain binomial products have special forms. When a binomial is squared, the result is called a perfect square trinomial. We can find the square by multiplying the binomial by itself. However, there is a special form that each of these perfect square trinomials takes, and memorizing the form makes squaring binomials much easier and faster. Let’s look at a few perfect square trinomials to familiarize ourselves with the form.

$(x+8)^2$

$\begin{array}{rrrll} x+8 \\ x+8 \\ \hline{8x+64} \\ x^2+8x \qquad\\ \hline{x^2+16x+64} \end {array}$

Or let’s look at $(y-5)^2$

	$y$	$-5$
$y$	$y^2$	$-5y$
$-5$	$-5y$	$25$

$y^2-10y+25$

And $(3x+2)^2=(3x+2)(3x+2)=9x^2+6x+6x+4=9x^2+12x+4$ .

Do you see the pattern? $(First + Last)^2=First^2+ 2\cdot First \cdot Last+Last^2$

Perfect Square Trinomial

When a binomial is squared, the result is the first term squared added to double the product of both terms and the last term squared.

$\left( a+b \right)^2= (a+b)(a+b)=a^2+2ab+b^2$

Given a binomial, square it using the formula for perfect square trinomials.

Square the first term of the binomial.
Square the last term of the binomial.
For the middle term of the trinomial, double the product of the two terms.
Add and simplify.

Warning Triangle with Exclamation Point

Don’t forget that middle term! Remember that $\color{red} (a+b)^2 \neq a^2+b^2$

Examples of expanding perfect square trinomials

$\left(3x-8 \right)^2$

Square $(3x)$ and remember to square them both: $9x^2$
Square $(-8)$ which is $64$
Now take $2 \cdot 3x \cdot -8$ , which is $-48x$
Add and simplify: $9x^2-48x+64$

Try in Now 5

$\left(2x+5 \right)^2$

Difference of Squares

Another special product is called the difference of squares, which occurs when we multiply a binomial by another binomial with the same terms but the opposite sign. Let’s see what happens when we multiply $(x+2)(x-2)$ .

$\begin{array}{rrrll} x+2 \\ x-2 \\ \hline{-2x-4} \\ x^2+2x \qquad\\ \hline{x^2+0x-4} \end {array}$

Note that we ended up with $0x = 0$ so $x^2-4$ .

Whenever we multiply anything of this pattern, the $x$ -term drops out and we end up with the difference of two squares.

Difference of Two Squares

When a binomial is multiplied by a binomial with the same terms separated by the opposite sign, the result is the square of the first term minus the square of the last term.

$(a+b)(a-b) = a^2-b^2$