FACTOR THEOREM SYNTHETIC DIVISION Factor each One zero has been given 1) ƒ(x) = x³ ? 6x² ? 15x+100; 5 3) ƒ(x) = x³ ? x² ? 8x + 12; ?3
use long and synthetic division to divide polynomials • use the remainder theorem • use the factor theorem Example 1: Use long division to find the
We will use synthetic division and the Remainder Theorem to do this Step 1: Use synthetic division to divide P(x) = 2x 3 + 5x
Factor Theorem: A polynomial f(x) has a factor x - k if and only if f(k) = 0 Example: Divide x2 - 5x + 6 by x - 2 2 1 -5 6 + 2 -6
1 Use synthetic division to perform the following polynomial divisions Find the quotient and the remainder polynomials, then write the dividend, quotient and
The leading coefficient of f(x)=4x5 + 3x2 ? 1 is 4 Rational Zero's Theorem Theorem The set composed of every factor of the constant term of a polynomial f(x)
Factor f x( ) completely, and find all of its real zeros Solution We will use Synthetic Division to show that 2 is a zero: By the Remainder Theorem, f 2( )=
Divide Use synthetic division Show all work neatly on another sheet of paper 2 3 Real Zeros of Polynomial Functions: Remainder Theorem
Use long division to divide, and use the result to factor the polynomial completely The remainder theorem tells that synthetic division can be used to
The Factor Theorem – A polynomial f(x) has a factor (x – k) if and only if ( ) 0 f k = Using the Remainder in Synthetic Division – The remainder r,
factors of polynomials Of the things The Factor Theorem tells us, the most pragmatic is that we had better find a more efficient way to divide polynomials by
Method 1 Synthetic Substitution By the Remainder Theorem, f(–1) should be the remainder when you divide the polynomial by x + 1 –1 5 2 –1 4 –5 3 –2 5