Add Subtract And Multiply Polynomials

Subtracting Polynomials

In subtraction, let me show you the underlining method.

(3x² - 2x + 5) - (2x² - 6x + 7)

First, I am going to distribute the -1 into the second expression. That will make this an addition problem!

The first expression stays the same: 3x² - 2x + 5. We will distribute the negative, like this: -1 × 2x², which is -2x²; -1 × -6x, which is a positive 6x; and -1 * 7, which is -7. This gives us our new subtraction problem: (3x² - 2x + 5) + (-2x² + 6x - 7).

Remember, I am going to show you the method of underlining instead of circling to add the expressions.

We look at the first term 3x^2 and underline it. Now, I continue to look for a like term. Here it is, -2x², and I underline it. Now I add them: 3x² + (-2x²), and we get x². That's going to be the first term of our answer.

The second term is -2x, and this time, I put a squiggly line under it. Now I continue to look for a like term. Here it is: 6x, and I put a squiggly line under that one. I add -2x + 6x, and I get 4x. That's the second term in our answer.

The last term is 5, and this time I put a jagged line under it. Now, I continue to look for a like term. Here it is: -7, and I underline it. Now, I add them: 5 + -7, and we get -2. This will be my last term in the answer: x² + 4x -2.

Multiplying Polynomials

Now, we get to multiplication. This problem won't quite work like addition or subtraction, and we can't use FOIL because these are larger than a binomial times a binomial!

(x+5)( x²+3x-2)

First, multiply the first term in the product: x times everything in the second expression.

I like to draw arrows to remind me which multiplication I've done; otherwise I tend to get lost.

This is how it will look:

x(x²) + x(3x) + x(-2)

Let's multiply.

x(x²) = x³ + x(3x) = 3x² + x(-2)= -2x.

This isn't our final answer; we need to multiply everything in the second expression by 5!

So we'll have 5(x²) + 5(3x) + 5(-2)

Are you ready for the final answer? We simply add the like terms together!

x³ + 3x² - 2x + 5x² + 15x - 10

Start from the left, and circle x³. It looks like there aren't any like terms for x³, so we write that down as our final answer.

Put a square around 3x². I look and find 5x², so I put a square around that term, too. I don't see any more, so 3x² + 5x² = 8x². 8x² is written next to x³ as part of our final answer.

Put a triangle around -2x. I look and find 15x, so I put a triangle around that term too. Why? Well, they're like terms. I don't see any more like terms for -x, so -2x + 15x = 13x. 13x is part of our final answer, and I'm going to write it next to 8x².

Finally, I see -10. I underline this term and look for another one like it. I don't see one, so -10 is written in my final answer.

So, what is the final answer then?

x³ + 8x² + 13x - 10.