RWM102: Algebra

8a. Identify the degree of a polynomial

• How do you identify the degree of a polynomial?

Polynomials are often classified by their degree, which is the highest power of any of the terms in the polynomial. For example, in the polynomial $y=5x^3-4x+7$, the highest power on the variable is 3, therefore this is considered a 3rd degree polynomial. The degree is always the highest power on the variable, even if that isn't the first term, in standard form.

To review, see Introduction to Polynomials. 

8b. Classify polynomial according to the number of terms 

• What do you call a polynomial?

We can also classify polynomials by their number of terms. For example, a single term, like $2x^3$ is called a monomial. Two monomials added or subtracted together form a binomial, and three monomials together are called a trinomial. Each of those monomials is called a term. If there are more than 3 terms, we generally just call it a polynomial.

For review, see Introduction to Polynomials. 

8c. Add, subtract, and multiply polynomials

• How do you add polynomials?
• How do you subtract polynomials?
• How do you multiply polynomials?

When adding polynomials, we will follow the rules we already know about collecting like terms. Like terms must have the same variable and exponents in order to add them, and the same is true for polynomials. For example, to add the polynomials $3x^4-5x^2+x+5$ and $2x^4+x^3+4x+3$, we identify the like terms, and add their coefficients.

$3x^4+2x^4=5x^4, x+4x=5x$, and $5+3=8$

Note that $-5x^2$ and $x^3$ have no like terms and therefore won't be added to anything. Now we can write our final answer, $5x^4+x3-5x^2+5x+8$.

When subtracting two polynomials, we can simply apply the subtraction to each term in the second polynomial and then add as we just did. For example, $(3x^3+2x^2+1)-(x^3-3x^2+5)$ can be rewritten as $(3x^3+2x^2+1)+(-x^3+3x^2-5)$. Now, this problem is the same as an addition problem, and we can simply collect the like terms.

Multiplying polynomials is a bit more work, and the approach depends somewhat on the types of polynomials being multiplied. First, let's consider a monomial multiplied by any other polynomial, such as this:

$3x(5x^2-3x+1)$. To simplify, we need to apply the distributive property, which states that we can multiply the $3x$ times each term inside the parentheses individually:

$3x(5x^2-3x+1)=(3x)(5x^2)+(3x)(-3x)+(3x)(1)=15x^3-9x^2+3x$

If we multiply anything other than a monomial by our polynomial, we have to use a slightly more complex version of the distributive property, where every term in the first polynomial is individually multiplied by every term in the second polynomial, then all like terms are collected and the final answer is found. For example:

$(x+2)(x^2-4x-1)=(x)(x^2)+(x)(-4x)+(x)(-1)+2(x^2)+2(-4x)+2(-1)$.

Now, as we multiply, we must also take note that there are like terms that will need to be collected:

$x^3-4x^2-x+2x^2-8x-2$

Once we collect our like terms, we get our final answer: $x3-2x^2-9x-2$.

To review, see:

• Adding and Subtracting Polynomials
• More on Adding and Subtracting Polynomials
• Multiplying Monomials by Polynomials
• Using the FOIL Technique to Multiply Binomials
• Multiplying Binomials by a Binomial
• Multiplying a Polynomial by a Binomial
• Multiplying Polynomials

8d. Divide a polynomial by a monomial and a binomial

• How do you divide a polynomial by a monomial?
• How do you divide a polynomial by a binomial?

Dividing polynomials requires us to carefully consider the type of polynomial that we are dividing by.

First, let's consider dividing by a monomial. When we divide a polynomial by a monomial, we can simply divide each term of the polynomial in the numerator by the monomial in the denominator. Each of these are then simplified to get our final answer. For example, consider $\dfrac{3x^4+2x^2-6}{3x^2}$. We first rewrite this problem so each term in the numerator is divided individually, then we simplify each of those fractions as such:

$\dfrac{3x^4}{3x^2}+\dfrac{2x^2}{3x^2}-\dfrac{6}{3x^2}=x^2+\dfrac{2}{3}-\dfrac{2}{x^2}$

Dividing a polynomial by a binomial is a bit more challenging. We will use long division, similar to what you learned when you first learned how to divide by hand, to divide these polynomials. Consider the following example:

When dividing a polynomial by a binomial, you simply need to follow this division algorithm until you finish the problem.

To review, see: 

• Dividing Polynomials with Monomials with Remainder
• Dividing Polynomials
• Dividing Polynomials by Binomials

8e. Identify special products of binomials by completing the square and finding the difference of two squares

• How do you square a binomial?
• How do you multiply a difference of squares?

Although by now we are comfortable with exponents, an exponent on a binomial can be computed incorrectly without the correct approach. When squaring a binomial, you cannot simply square each term in the binomial. You must write the binomial twice, and use the distributive property to multiply each term in each binomial with the terms in the other binomial.

For example: $(x+3)^2=(x+3)(x+3)$. Now, we multiply as we have learned, and collect the like terms:

$(x+3)(x+3)=x^2+3x+3+9=x^2+6x+9$

This is the correct way to multiply binomials. Be careful not to forget this step.

Another special kind of binomial multiplication is called a difference of squares. A difference of squares is when the two binomials being multiplied are of the form $(x+a)(x-a)$. When multiplied, the inner and outer terms always cancel, giving an answer of $x^2-a^2$. For example, $(x+5)(x-5)=x^2-25$.

To review, see Finding the Square of a Binomial with One Variable and Special Products of Polynomials. 

Unit 8 Vocabulary

This vocabulary list includes terms listed above that students need to know to successfully complete the final exam for the course.

• polynomials
• highest power
• degree
• monomial
• binomial 
• trinomial 
• like terms
• coefficients 
• long division
• division algorithm
• squaring
• difference of squares