Factoring a number means writing it as the product of other numbers. Can the same thing be done to polynomials? Absolutely! Although polynomials may look quite complicated, polynomials represent numbers just like a variable represents a number. One huge idea in algebra is that anything we can do to a number, we can probably do to a polynomial.
Number example: 30 = 5(6)
In essence, a polynomial is factored by “extracting” all the factors shared by each of the polynomial’s pieces. The result is an equivalent expression, except this time it’s written as the product of two shorter polynomials. How do we check our work? Use the distributive property. So what’s another name for factoring? The reverse distributive property!
The main idea is that two entire polynomials can be multiplied with the distributive property. Example:
Find the product of and
Instead of distributing a single term, we can distributive an entire polynomial. Whoa! In this problem, we can distribute the :
Let’s reflect on what just went down. The entire polynomial was distributed to the “x” in and then to the “3” in .
OK, now what? We are going to use the distributive property again!
Maybe you’ll encounter some people who try to teach this topic using all sorts of tips and tricks. That’s OK. But never lose sight of the fact that multiplying polynomials is nothing other than the distributive property!
The distributive property says there are always two ways of counting something: ALL AT ONCE or PIECE BY PIECE.
For example, if you have 3(2x + 6), then it’s also truthful to say that you’ve got three 2x’s and three 6’s. In other words, 3(2x + 6) = 3(2x) + 3(6). The purpose of today’s lesson was to apply the distributive property to variables. Note that this calls upon our old ability to multiply monomials like (2xy)(3x).
The main purpose of today’s lesson was to review adding and subtracting polynomials, but in a more difficult context.
- If “A is subtracted from B”, then the proper subtracting problem is B – A. For example, if (x – 7) is subtracted from (x + 10), then the resulting difference would be (x + 10) – (x – 7) = 17.
- If a negative sign is in front of a parenthesis, the effect is to take the opposite of every term inside. For example, if a problem says “Simplify: -(-2x + 9)”. The answer would be 2x – 9.
The focus of today’s lesson was on combining like terms.
Essentially, combining like terms is a fancy form of counting. For example, if you have 3 sheep and you also have 5 sheep, you have 8 sheep in total. Likewise, if you have 3 ‘s (“x squared’s”) and you also have 5 ‘s, you have 8 ‘s in total.
When you add or subtract like terms, the exponents stay the same and you simply add or subtract the coefficients.
Dividing things with coefficients and variables (called monomials or terms) is almost exactly the same thing as multiplying. First, we can rewrite each term the long way using the fact that exponents represent repeated multiplication. Then we use the fact that when a number or variable is divided by itself, they “cancel” each other out to be one.
In short, when you divide, you should divide the coefficients and subtract the exponents.
In contrast, when you multiply, you should multiply the coefficients and add the exponents.
The main idea of today’s lesson is that multiplication of a number or variable by itself multiple times can be rewritten using exponents.
Based on this main idea, there are three rules that can be established:
- Multiply the coefficients
- Add the exponents
- If a variable has no exponent, the exponent is actually 1