1. 5.3: Add, Subtract, & Multiply Polynomials
Objectives:
To define a polynomial and related terms
To find the sum, difference, and product of polynomials

2. Polynomial What makes one of these things a polynomial? Polynomial
NOT a Polynomial
2x2
3xy
x4 – 81
2x - √x
x2 – 3x + 4
x-1 + 2x-2 – 4x-3

3. Polynomial A polynomial in x is an expression of the form: Monomial
Binomial
Trinomial
One term
Two terms
Three terms
2x2
x4 – 81
x2 – 3x + 4

4. Like Terms
Like terms are simply monomials with the same variable raised to the same power
To add and subtract polynomials, just add or subtract the coefficients of like terms
The powers DO NOT change!

5. Exercise 1
Add using a horizontal format. It’s often helpful to underline like terms the same number of times.

6. Exercise 2
Add using a vertical format. Align like terms and add the old-fashioned way.

7. Distributive Property
Distributive Property When subtracting polynomials, you have to distribute the negative sign:

8. Exercise 3
Subtract 6y2 – 6y – 13 from 3y2 – 4y + 7 in a horizontal format.

9. Exercise 4
Subtract –4x3 + 6x2 + 9x – 3 from 3x3 + 4x2 + 7x + 12 in a vertical format. Basically you have to turn a subtraction problem into addition by adding the opposite.

10. Exercise 5 Find the sum or difference. (t2 – 6t + 2) + (5t2 – t – 8)
(8d – 3 + 9d3) – (d3 – 13d2 – 4)

11. Polynomial Multiplication
Polynomial multiplication is basically repeated application of the distributive property. Multiply coefficients and add exponents

12. Polynomial Multiplication
Polynomial multiplication is basically repeated application of the distributive property.
Product of the First terms
Product of the Outside terms
Product of the Inside terms
Product of the Last terms

13. Polynomial Multiplication
Polynomial multiplication is basically repeated application of the distributive property.
First
Outside
Inside
Last

14. Exercise 6
Find the product. Difference of two squares

15. Protip: Difference of 2 Squares
When finding the difference of 2 squares, just square the first number, square the second number, and take the difference. The middle term cancels out.
Square the first term
Square the second term

16. Exercise 8
(5y – 3)(5y + 3)
(4a + 7)2
(2x – 3)2

17. Polynomial Multiplication
When multiplying a polynomial by a polynomial, each term of the first polynomial must be multiplied by each term of the second polynomial.
Again, this is just the distributive property used multiple times.

18. Exercise 9
Multiply x2 – 2x + 3 and x + 5 in a horizontal format.

19. Exercise 9
Multiply x2 – 2x + 3 and x + 5 in a horizontal format.

20. Exercise 10
Multiply 3x2 + 3x + 5 and 2x + 3 in a vertical format.

21. Exercise 10
Multiply 3x2 + 3x + 5 and 2x + 3 in a vertical format.

22. Exercise 11 Find the product. (x + 2)(3x2 – x – 5)

23. Exercise 14
Find the product. (a – 5)(a + 2)(a + 6)