2.01 Adding and subtracting polynomials
What is a polynomial?
A polynomial is a mathematical expression with many terms ("poly" means "many" and "nomial" means "names" or "terms").
A polynomial can have any combination of operators (addition, subtraction, multiplication or division), constants, variables and exponents, but never division by a variable. Remember, this means that expressions with negative exponents can never be polynomials because $a^{-x}$a−x is the same as $\frac{1}{a^x}$1ax.
Polynomials are usually written in descending order, starting with the term with the greatest power and ending with the term with the least (or no) power. For example, in the polynomial $8x+4x^8-2x^2+4$8x+4x8−2x2+4, the powers are all jumbled. To put it in order, we would rewrite it as $4x^8-2x^2+8x+4$4x8−2x2+8x+4.
| Examples of expressions that ARE polynomials | Examples of expressions that ARE NOT polynomials |
|---|---|
| $5x^2+\frac{4}{3}x-7$5x2+43x−7 | $\frac{4}{x-3}$4x−3 |
| $-18$−18 | $3+\frac{1}{x}$3+1x |
| $3x$3x | $4x^3-\frac{1}{x^7}+8$4x3−1x7+8 |
| $4c-8cd+2$4c−8cd+2 | $\frac{7}{8}x^{-2}+5$78x−2+5 |
| $22x^6+12y^8$22x6+12y8 | $\sqrt{x}$√x |
| $7g+\sqrt{12}$7g+√12 | $12f^3g^{-4}\times h^6$12f3g−4×h6 |
If an expression contains terms that divide by a variable (i.e. an algebraic term), they are NOT polynomials.
Parts of a polynomial
Degree: The largest exponent or the largest sum of exponents of a term within a polynomial. For example, in the polynomial $x^3+4x^2-9$x3+4x2−9, the greatest power of $x$x is $3$3, so the degree in this polynomial is $3$3.
Leading coefficient: The coefficient of the first term of a polynomial written in descending order of exponents. For example, in $5x-7$5x−7, the leading coefficient is $5$5 and in $-x^5-2x^4+4$−x5−2x4+4, the leading coefficient is $-1$−1.
Constant term: the term in a polynomial that has no variables (degree of 0). For example, in the polynomial $4y^8+2xy-4x-\frac{2}{3}$4y8+2xy−4x−23, the constant term is $-\frac{2}{3}$−23.
Types of polynomials
- A linear polynomial is the equation of a straight line, which can be written in the form $f\left(x\right)=mx+b$f(x)=mx+b.
- A quadratic polynomial is the equation of a parabola, which can be written in the form $f\left(x\right)=ax^2+bx+c$f(x)=ax2+bx+c.
- A cubic polynomial, which can be written in the form $f\left(x\right)=ax^3+bx^2+cx+d$f(x)=ax3+bx2+cx+d.
- A monomial has a single term, for example, $f\left(x\right)=7x$f(x)=7x
- A binomial has two terms, for example, $f\left(x\right)=3x^2+8$f(x)=3x2+8
- A trinomial has three terms, for example, $f\left(x\right)=3x^2-5x+11$f(x)=3x2−5x+11
Practice questions
Question 1
Is $2x^3-4x^5+3$2x3−4x5+3 a polynomial?
Yes, it is a polynomial.
ANo, it is not a polynomial.
B
Question 2
For the polynomial $P\left(x\right)=\frac{x^7}{5}+\frac{x^6}{6}+5$P(x)=x75+x66+5
The degree of the polynomial is: $\editable{}$
The leading coefficient of the polynomial is: $\editable{}$
The constant term of the polynomial is: $\editable{}$
Adding and subtracting polynomials
Adding and subtracting polynomials is very similar to the process of simplifying algebraic expressions by collecting like terms.
Just like with any algebraic expression, we can only add and subtract like terms.
The process
Let's run through the process by looking at an example. Let's say we want to find the difference between two polynomials: $P\left(x\right)=7x^3+4x^2-4$P(x)=7x3+4x2−4 and $Q\left(x\right)=7x^3+8x^2-2x-8$Q(x)=7x3+8x2−2x−8.
1. Start by writing out the equation we want to simplify:
$P\left(x\right)-Q\left(x\right)=7x^3+4x^2-4-\left(7x^3+8x^2-2x-8\right)$P(x)−Q(x)=7x3+4x2−4−(7x3+8x2−2x−8)
2. Collect the like terms, taking any negative symbols into account.
Don't forget to subtract the entire polynomial. We must write the polynomial being subtracted in parentheses to remind us that the negative applies to every term being subtracted.
Remember if there is a term with no corresponding term in the other polynomial, we can treat this as a value of zero. For example, $P\left(x\right)$P(x) does not have a term with $x$x but $Q\left(x\right)$Q(x) does.
We can present it vertically in a table and align like terms:
| $7x^3$7x3 | $+$+ | $4x^2$4x2 | $+$+ | $0$0 | $-$− | $4$4 | |||
| $-$− |
$($( |
$7x^3$7x3 | $+$+ | $8x^2$8x2 | $-$− | $2x$2x | $-$− | $8$8 |
$)$) |
| $0$0 | $-$− | $4x^2$4x2 | $+$+ | $2x$2x | $+$+ | $4$4 |
or we can simply distribute the negative and group the like terms:
| $7x^3+4x^2-4-\left(7x^3+8x^2-2x-8\right)$7x3+4x2−4−(7x3+8x2−2x−8) | $=$= | $7x^3+4x^2-4-7x^3-8x^2+2x+8$7x3+4x2−4−7x3−8x2+2x+8 |
| $=$= | $7x^3-7x^3+4x^2-8x^2+2x-4+8$7x3−7x3+4x2−8x2+2x−4+8 | |
| $=$= | $-4x^2+2x-4$−4x2+2x−4 |
You can choose what method you like.
3. Write out the solution
$P\left(x\right)-Q\left(x\right)=-4x^2+2x+4$P(x)−Q(x)=−4x2+2x+4
Practice questions
Question 3
Simplify $\left(3x^3-9x^2-8x-7\right)+\left(-7x^3-9x\right)$(3x3−9x2−8x−7)+(−7x3−9x).
Question 4
If a picture frame has a length of $8x^2-9x+3$8x2−9x+3 and a width of $6x^3+9x^2$6x3+9x2, form a fully simplified expression for the perimeter of the rectangular picture frame.
Question 5
If $P\left(x\right)=3x^2+7x-6$P(x)=3x2+7x−6 and $Q\left(x\right)=6x-7$Q(x)=6x−7, form a simplified expression for $P\left(x\right)-Q\left(x\right)$P(x)−Q(x).