Synthetic Division
***Only use with linear divisors***
***Use “0” for missing terms****
Uses other than division:
1) To test factors or roots/zeros/solutions:
Synthetic division for a value of “x” with a remainder = 0
indicates that it IS a factor/root/zero/solution.
2) To find missing factors/roots/zeros/solutions:
Divide by known or given roots in order to factor or solve remaining polynomial.
3) Remainder Theorem:
INSTEAD of substituting a value of “x” into the polynomial,
the REMAINDER = VALUE.
DIVISION OF POLYNOMIALS
Synthetic Division
***Only use with linear divisors***
***Use “0” for missing terms****
***Value of “x” goes in “box”***
***Bring down 1st coefficient, multiply, add***
***Each division reduces degree by 1***
***Numbers represent new coefficients***
Long Division
***Use with ANY divisor***
***Use “0x” for any missing terms***
***Same process as for numbers: Divide, Multiply, Subtract, Bring down***
Finding Polynomials
***Use roots/zeros to write factors***
***Always have OPPOSITE of irrational and imaginary roots too***
***Multiply factors and combine like terms***
***Write in standard form***
Binomial Expansion
***use instead of multiplying binomial factor by itself repeatedly***
***Use row of numbers from Pascal’s Triangle with descending powers of first term and ascending powers of last term***
Graphing Polynomials
End Behavior
***Describes the direction of the right and left ends of a polynomial function***
***Depends of if degree is EVEN (both same direction) or ODD (opposite directions)
and
sign of “a” (+ right side up,
-- right side down)***
Roots/ Zeros
***Real roots/zeros represent X-INTERCEPTS***
Solving/Finding roots or zeros
By factoring:
***Check for and factor GCF***
***Technique depends on number of terms***
2 terms: formula
sum of cubes : (a+b)(a2-ab+b2)
difference of cubes: (a-b)(a2+ab+b2)
difference of squares: a+ba-b
***no sum of squares***
3 terms: guess and check
4 terms: grouping
***set each factor = 0 and solve***
Rational Root Theorem
***use when you can’t factor***
***possible roots =±factors of the last number±factors of "a"
***divide by roots found in calculator until
degree= 1 or 2***
***solve remaining polynomial***
Other Uses of Synthetic Division
1. Is (x-3) a factor of the polynomial 2x3+4x2-10x-9?
(Show work to justify your answer.) Explain how you can tell.
2. Use the remainder theorem to evaluate
2x3-4x2+10x+5 if x = 12
3. If x = 2 is a root of x3-7x2+14x-8 , write the complete factorization of the polynomial.
4. Is 4i a zero of x4-3x3+6x2-48x-160? (Show work to justify your answer.)
Finding Polynomials
For each of the following, find the polynomial in standard form:
1. roots are x = 3, 13 , and -32
2. roots are x = 0 multiplicity of 2, and x = -1 multiplicty 2
3. roots are x = 3, 4
4. root is x = -2i +3
5. x+22-4x-x4+11
6. Use binomial expansion: (x+2)5
Solving/Finding Zeros/Roots
Solve each equation according to directions:
1. 3x3-81=0 by factoring
2. x4-18x2+32=0 by factoring
3. x3+6x2-5x-30 by factoring
4. x4-9x3+11x2-19x-40=0
5. If x=3-4i is a zero of x4-6x3+29x2-24x+100, find all other zeros.
Khan Academy Practices:
“Graphs of Polynomials”- Use roots and end behavior to match graphs to polynomial equations.
“Dividing Polynomials with remainders”-
Use synthetic or long division to divide. Remainders should be entered as a fraction with +/- sign in front.