7.3 Integral & Rational Zeros of Polynomial Functions

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

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)

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 We can now use depressed equation of 6x2 + 7x + 2 = 0 & solve Factor or quadratic? Either!

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

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 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