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## SAT (Fall 2023)

### Course: SAT (Fall 2023) > Unit 10

Lesson 2: Passport to advanced mathematics- Solving quadratic equations — Basic example
- Solving quadratic equations — Harder example
- Interpreting nonlinear expressions — Basic example
- Interpreting nonlinear expressions — Harder example
- Quadratic and exponential word problems — Basic example
- Quadratic and exponential word problems — Harder example
- Manipulating quadratic and exponential expressions — Basic example
- Manipulating quadratic and exponential expressions — Harder example
- Radicals and rational exponents — Basic example
- Radicals and rational exponents — Harder example
- Radical and rational equations — Basic example
- Radical and rational equations — Harder example
- Operations with rational expressions — Basic example
- Operations with rational expressions — Harder example
- Operations with polynomials — Basic example
- Operations with polynomials — Harder example
- Polynomial factors and graphs — Basic example
- Polynomial factors and graphs — Harder example
- Nonlinear equation graphs — Basic example
- Nonlinear equation graphs — Harder example
- Linear and quadratic systems — Basic example
- Linear and quadratic systems — Harder example
- Structure in expressions — Basic example
- Structure in expressions — Harder example
- Isolating quantities — Basic example
- Isolating quantities — Harder example
- Function notation — Basic example
- Function notation — Harder example

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# Operations with polynomials — Basic example

Watch Sal work through a basic Operations with polynomials problem.

## Want to join the conversation?

- dude i thought thay said BASIC
**example**lol(10 votes) - So just double checking, at2:04, the -2 comes because the minus sign in front of the equation at the part before?(8 votes)
- Just wondering - How many colors does he have?(6 votes)
- how does 1 + 1/2 =3/2(3 votes)
- You can think of this as adding two fractions with unlike denominators, 1/1 and 1/2. In order to do this, we have to change the denominator of one of the function to match the other, so that we can add the numerators to each other and keep the denominator the same. We can change the denominator of a fraction by multiplying the numerator and denominator by the same number, and then add across:

1 + 1/2

= 1/1 + 1/2

= (1*2)/(1*2) + 1/2

= 2/2 + 1/2

= (2+1)/2

= 3/2

(For a more detailed explanation of this, Khan has a nice video on this exact topic)(5 votes)

- 2:03why did he start subtracting?(4 votes)
- Because when he simplified the parenthesis(by multiplying everything inside) all the terms in the parenthesis became negative. That is why he is subtracting.(2 votes)

- Wait why isn't it -1/44 since 1/88y^100 and -1/44y^100 are common factors? Where did multiplying the 2 with 1/44 come into play?(3 votes)
- Yes, 1/88 y^100 and -1/44 y^100 are like terms. You multiply the numerator and denominator of -1/44 by 2 to be able to add it to the 1/88. 1/44 is twice 1/88, so when you subtract it from 1/88, you get the negative of 1/88. Does this make it more clear?(2 votes)

- so, considering the fact that someone I know has their psat coming up, can anyone explain to me the different ways of solving polynomials. (Addition, subtraction, etc)(2 votes)
- When you add polynomials, all you do is add the like terms. You can't add an x term with an x^3 term, for example.

Ex: (5k^3 + 3k^2) + (6k^3 + 5mk^2 + 12m)

The only like terms here are 5k^3 and 6k^3, so they are the only terms you'll combine:

= (6 + 5)k^3 + 3k^2 + 5mk^2 + 12m

= 11k^3 + 3k^2 + 5mk^2 + 12m

Subtracting polynomials is the same process as adding, but with one additional step. Since subtracting is the same as adding the negative, make every term in the polynomial you're subtracting the opposite. Then add as normal.

To multiply polynomials, you multiply every possible pair of polynomials, and add those products up. When you multiply, you multiply the coefficients and add exponents if they have the same base.

Ex: (k^2 + 5) * (5m^4 - k^2 + 9)

= (k^2 * 5m^4) + (k^2 * -k^2) + (k^2 * 9) + (5 * 5m^4) +(5*-k^2)+(5*9)

= 5m^4k^2 - k^4 + 9k^2 + 25m^4 - 5k^2 + 45

= 25m^4 + 5m^4k^2 - k^4 +4k^2 + 45

Polynomial division won't be directly tested on the SAT. If you're curious, you can check out this Khan Academy playlist: https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-div(2 votes)

- How did you get 1 + 1/2 to equal 3/2. I thought it equal 2/2.(1 vote)
- You can't add the numerators of fractions unless their denominators are the same. Then, to add fractions, the numerator of the sum is the sum of the 2 numerators and the denominator of the sum is the same.

Here, we can think of whole numbers as having a denominator of 1. So 1 would be 1/1. Because of this, you cannot add it to a fraction with denominator 2. Instead, we have to convert it by multiplying both the numerator and denominator by the same constant (in this case 2).

1 + 1/2 = 1/1 + 1/2

= ((1 * 2) / (1 * 2)) + 1/2

= 2/2 + 1/2

= 3/2

Does this clear it up for you?(3 votes)

- In0:56he put 2k^2 but why I thought it was just 2k?(2 votes)
- Sal was using the FOIL method to multiply the two binomials (first, outer, inner, last), so he was multiplying the first term of (k+1), or "k", by the first term of (2k+2), or "2k".

k•2k=k•2•k=2•k•k=2•(k^2)=2k^2.(2 votes)

- Are Polynomials in IGCSE Add Math the same as the one in SAT Math and does this apply to the rest of the topics in IGCSE Add Math.(1 vote)

## Video transcript

- [Instructor] Which of
the following is equivalent to all of this stuff right over here. One over 88, or 1/88,
y to the 100 plus one, minus this entire expression, 1/44 y to the 100 minus 1/2. So let me just rewrite it. So this part right over here, this is going to just be one over 88 y, let me make sure it looks clear, y to the 100th power plus one, and then we're gonna subtract all of this. So one way to think about
it is we could distribute this negative sign, so it's gonna be, it's going to be negative, one way you could think
about it is negative one, negative one times all of this business. So it's gonna be minus one
over 44 y to the 100th, and then a negative, remember,
we're gonna distribute it, so negative one times negative 1/2 is going to be plus, plus 1/2. Now, let's see what we can
do to further simplify it. So this is 1/88 y to the 100. This is minus, or you could say negative, 1/44 y to the 100th. And so we can add these
two terms together. This is going to be, we could write this, one over 88 minus one over 44 y to the 100th, and then you have plus one, plus 1/2, so that's pretty straightforward. So you have plus one, plus 1/2, so that's gonna be plus 3/2, plus 3/2. And now we just have to
figure out what this is. So 1/88 minus 1/44, well,
the common denominator here could be 88. That is the least common
multiple of 44 and 88. 88 is divisible by 44. So if we multiply the
denominator by two, you get 88. You multiply the numerator
by two, you get two. 2/88 is the same thing as 1/44. So this 1/88 minus 2/88. Well, this is going to give
us, this right over here is going to give us, if we, this is going to give us
minus, or negative, 1/88. 1/88 minus 2/88 is going
to be negative 1/88, y to the 100, y to the 100, plus 3/2, plus 3/2. And that is negative
1/88th y to the 100th, y to the 100th plus 3/2. This choice right over here.