7.3 Integral & Rational Zeros of Polynomial Functions

1. 7.3 Integral & Rational Zeros of Polynomial Functions

2. Let’s start by factoring two easy polynomials and make an observation about the factors *What do you notice about the last number and each of the factors? *This is not a coincidence! We can obtain a list of possible zeros for an equation by taking factors of the leading coefficient and the constant Rational Zeros Theorem A number can be a rational zero of a polynomial only if it is of the form where p is a factor of the constant and q is a factor of the leading coefficient

3. Ex 1) Determine the possible rational zeros of each polynomial What if they asked for just the possible integral zeros? (this means just the integers) *go back & circle just the integer answers on a) & b)

4. So, now that we have possible choices, we can narrow down what to try when we actually find the zeros or factor a polynomial *Note: our calculators can graph & also guide us in finding them! Ex 2) Determine the zeros of each polynomial Consult graph … looks like possibly 1 … try it! 1 6 1 –5 –2 ↓ 6 7 2 6 7 2 0 We can now use depressed equation of 6x2 + 7x + 2 = 0 & solve Factor or quadratic? Either!

5. Ex 2) cont… 6x2 + 7x + 2 = 0 Consult graph … try –3 –3 1 8 17 6 ↓ –3 –15 –6 1 5 2 0 x2 + 5x + 2 = 0

6. Application: Making a box Ex 3) Open-top boxes are being made from a 10 in. × 13 in. sheet of cardboard by cutting out small squares from the corners and need to have a volume of 88 in3. What size square should be cut out to get the desired volume? 1 4 –46 130 –88 x 13 x ↓ 4 –42 88 x x 4 –42 88 0 10 4x2 – 42x + 88 = 0 x x x x V = l • w • h = (13 – 2x)(10 – 2x)(x) = (130 – 46x + 4x2)(x) = 130x – 46x2 + 4x3 0 = 4x3 – 46x2 + 130x – 88 makes dimensions (–) 88 cut 1 in. or 2.89 in. squares out

7. Homework #703 Pg 347 #4, 9, 10, 15, 17, 24, 26–28, 32, 34, 37, 38–41