Real Zeros of Polynomials

Section 2.4

Review – Long Division

1. What do I multiply by to get the first term?

2. Multiply through

3. Subtract

4. Bring down next term

CONCLUSION

CONCLUSION

Review: Synthetic Division

1. Solve the divisor for “x”

2. Set up problem with coefficients ONLY

3. Bring down the first term

4. Multiply, add, repeat

FACTOR THEOREM

(x – k) is a factor of a polynomial if and only if f(k)

= 0

Practice Determine whether each binomial is a FACTOR

of f(x) = x3 + 3x2 – 6x – 8

1. (x – 2)

2. (x + 3)

3. (x + 4)

Rational Zeros Theorem

Rational #s – numbers that can be written as fractions

POSSIBLE rational zeros of a polynomial can

be found by dividing the factors of the LAST

TERM and LEADING COEFFICIENT

Practice – find possible rational zeros

f(x) = 3x3 + 4x2 – 5x – 2

Factors of -2:

Factors of 3:

1, -1, 2, -2

1, -1, 3, -3

1, -1, 2, -2

List all of the POSSIBLE rational zeros of

f(x) = 2x3 – x2 – 9x + 9

PUTTING IT ALL TOGETHER1. Find all of the zeros (rational & irrational)

f(x) = 2x3 – 3x2 – 4x + 6

Step 1: List possible rational zerosStep 2: Find one from the list that IS a zeroStep 3: Synthetic DivisionStep 4: Factor

Ticket Out1.) Is 2x + 1 a factor of 4x3 - 8x2 – 1 ? Show work

2.) Divide x3 – 5x2 + 3x – 2 by (x + 1) using LONG division

3.) Divide x3 – 5x2 + 3x – 2 by (x + 1) using SYNTHETIC

4.) List all possible rational zeros of f(x) = 2x4 + 3x – 3