Section 3 – The Remainder and Factor Theorems

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Presentation transcript:

Section 3 – The Remainder and Factor Theorems Chapter 4 Section 3 – The Remainder and Factor Theorems

Remainder Theorem The Remainder Theorem: is that the remainder of any polynomial divided by the binomial factor (x-r) gives the value of the polynomial at x=r. The Factor Theorem: is just a special case of this, which says if there is NO remainder, than (x-r) is a factor of the polynomial (since no remainder means that r is a zero).

Synthetic Division Synthetic Division is where we divide a polynomial by a binomial. EX 1: Divide x3 + 4x2 - 3x – 5 by x+3. The way we set this up is similar to long division with numbers. -3| 1 4 -3 -5 Start w/ root on left coefficients on right ONLY WORKS FOR linear binomial factors!

More EX 2: Use synthetic division to divide x3 – x2 + 2 by x+1. EX 3: Use synthetic division to find out if x-1 is a factor of 2x3 – 3x2 + x EX 4: Find the value of k so that the remainder of (x3 + 3x2 – kx – 24)/(x+3) is zero.

Finding Binomial Factors EX 5: Determine the binomial factors of x3 + x2 – 4x – 4 (this question is a lead-in to Rational Roots Theorem and Descartes’ Rule of Signs! Remember them???)

Assignment Chapter 4, Section 3 pgs 226-228 #12-36E,45