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

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

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

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!

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

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

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

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

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.

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

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

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

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

Exercise 6 Find the product. Difference of two squares

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

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

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.

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

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

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

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

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

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