Basic Algebra/Polynomials/Adding and Subtracting Polynomials
Polynomial: Mathematical sentence with "many terms" (literal English translation of polynomial). Terms are separated by either a plus (+) or a minus (-) sign. There will always be one more term than there are plus (+) or minus (-) signs. Also, the number of terms will (generally speaking) be one higher than the lead exponent.
EX: A Quadratic function has a lead exponent of 2, but generally has three terms (ax2 + bx + c; lead exponent = 2, # of + (or -) signs = 2, # of terms = 3)
Like Terms: Terms in a polynomial that have the same power of the variable
EX: 3x2 and 2x2 are like terms, but 3x2 and 4x3 are not!
PROPERTIES TO REMEMBER:
If we see a variable standing alone (it has no coefficient, no number next to it) then we assume that there is an invisible one (1) standing there:
x2 = 1x2
We are extremely lazy in math and do not like to write numbers that we feel are unnecessary. This is one of the cases in which we do not write what is actually there but we always remember it is. Another case is with fractions and whole numbers. The DEMONIMATOR of every whole number is a one (1) but we do not write this one (1) because we do not feel the need. We always remember it is there though.
4 = 4/1 [READ: 4 over 1]
There are many, many types of polynomials in the world of mathematics and are classified by the power (or exponent) of their leading term.
Some Common Functions
1) f(x) = ax + b (more commonly seen as y = mx + b). The leading term has an exponent of one (1) and is called a LINEAR FUNCTION because it creates a line when graphed. Because there are two terms, this function is called a binomial, or two-termed, function.
2) f(x) = ax2 + bx + c The leading term has an exponent of two (2) and is called a QUADRATIC FUNCTION because the first x is squared and squares are QUADrilaterals. This function generally has three terms and is therefore called a trinomial. A QUADRATIC FUNCTION has amazing properties that span years of mathematical studies. Since this is the first polynomial to have more than two terms, it is the first polynomial able to be factored. However, there are special cases in which ax2 + bx + c cannot be factored.
3) f(x) = ax3 + bx2 + cx + d The leading term has an exponent of three (3) and is called a CUBIC FUNCTION because the first x is cubed (raised to the third power). This function generally has four terms and will always be able to factor out at least one term of the form (x - h) [where h is any number].
There are an infinite number of polynomials and each one has amazing features unique to that function. However, there are a few universal traits to all functions. Every function with an even lead exponent (ax2, ax4, etc. . .) have a chance of not being factorable. Every function with an odd lead exponent (ax, ax3, etc. . .) will be able to factor AT LEAST ONE term of the form (x - h) [where h is any number].
As with regular numbers, we can add and subtract polynomials. However, instead of only worrying about which numbers have an x and which numbers do not, we also have to keep in mind that the exponents have to be the same in order for us to add and subtract terms.
EX: Add (4x3 + 3x + 1) + (-3x3 + 2x2 + 4)
Step 1: We have to match up our terms: (4x3 + -3x3) + (2x2) + (3x) + (1 + 4)
Step 2: We combine the coefficients of the like terms: x3 + 2x2 + 3x + 5 <- We've solved the problem
(4x3 + 3x + 1) + (-3x3 + 2x2 + 4) = x3 + 2x2 + 3x + 5
Subtracting polynomials is the same thing, except we add an extra step. When we subtract polynomials we use the distributive property first and multiply the second polynomial by a negative one (-1). This changes all the signs of the second polynomial to the OPPOSITE of what they are. [NOTE: When we add a negative number we actually subtract!!!]
EX: Subtract (3x4 + 2x2 + 2) - (x4 + 6x2 + 12x - 1)
Step 1: We distribute the negative one (-1) across the second polynomial and our new polynomial reads:
(-x4 - 6x2 - 12x + 1) <- Notice how the signs are all opposite of what we were given.
Step 2: We match up our terms: (3x4 + -x4) + (2x2 + -6x2) + (-12x) + (2 + 1)
Step 3: We combine our coefficients of the like terms: 2x4 - 4x2 - 12x + 3
(3x4 + 2x2 + 2) - (x4 + 6x2 + 12x - 1) = 2x4 - 4x2 - 12x + 3
Now we've successfully subtracted two polynomials.
1) (2x2 + 3x + 4) + (5x + x2 + 3) = ?? [= 3x2 + 8x + 7]
2) (-3x5 + 12x3 + 15x - 2) + (4x5 - 8x3 + 2x2 - 7x) = ?? [= x5 + 4x3 + 2x2 + 8x - 2]
3) (9x - 5) - (8x + 6) = ?? [= x - 11]
4) (3x3 + 2x) - (-x2 - 1) = ?? [= 3x3 + x2 + 2x + 1]