Transcript Academy Algebra II 5.5: FINISH: Polynomial Long Division and

Academy Algebra II
5.5: Polynomial Long Division
and Synthetic Division
HW: 5.4: p.357 (30-40 even, 46-54 even)
Divide
5 724
Polynomial Long Division vs.
Synthetic Division
• Polynomial Long Division can be
used for any two polynomials.
• Synthetic Division: the divisor
has to be in the form x – k,
where k is any constant.
Divide using polynomial long division:
2 x  5 6 x  11 x  26
2
Divide using synthetic division:
2 x
2
 7 x  10    x  5 
Divide: 2 x 3  11 x 2  13 x  44    x  5 


Divide: x 3  4 x  6   x  3 
Divide: 4 x  5 x  4    x  3 x  2 
4
2
5.5: Given polynomial f(x) and a
factor of f(x), factor f(x) completely.
f ( x )  x  10 x  19 x  30 ; x  6
3
2
Academy Algebra II
5.5: FINISH: Polynomial Long Division
and Synthetic Division
Hw: 5.5: p.366 (8, 22, 28, 30, 32, 36)
Given polynomial f(x) and a factor of
f(x), factor f(x) completely.
• Steps
1.) Divide the polynomial and the
factor.
2.) Factor the answer.
3.) Write out all factors.
Given polynomial f(x) and a factor
of f(x), factor f(x) completely.
f ( x )  2 x  15 x  34 x  21 ; x  1
3
2
Given polynomial f(x) and a zero
of f(x), find the other zeros.
• Zeros: answers to the polynomial
equation f(x) = 0.
• Process.
1.) Use the zero to factor the
polynomial completely.
2.) Solve to find the other zeros.
Given polynomial f(x) and a zero
of f(x), find the other zeros.
f ( x )  x  2 x  21 x  18 ;  3
3
2
Given polynomial f(x) and a zero
of f(x), find the other zeros.
f ( x )  10 x  89 x  12 x  27 ; 9
3
2
Given polynomial f(x) and a factor
of f(x), factor f(x) completely.
f ( x )  3 x  2 x  61 x  20 ; x  5
3
2
Given polynomial f(x) and a zero
of f(x), find the other zeros.
f ( x )  3 x  34 x  72 x  64 ;  4
3
2
Academy
Algebra II
5.6: Find Rational Zeros
HW tonight: p.374 (4-10 even)
Tomorrow: p.374 (14-20 even)
Next day: p.374-375 (24-30 even)
List all possible rational zeros using the
rational zero theorem.
• Every rational zero of a function has the
following form:
p
q

factor of constant t erm a 0
factor of leading
coefficien
t an
List all possible rational zeros using the
rational zero theorem.
• Example: List the possible rational zeros for the
function: f ( x )  x 3  2 x 2  11 x  12
Factors of the constant:  1,  2 ,  3,  4 ,  6 ,  12
Factors of the leading coefficient:  1
1
2
3
4
6
12
1
1
1
1
1
1
Possible rational zeros:  ,  ,  ,  ,  , 
Possible rational zeros:  1,  2 ,  3,  4 ,  6 ,  12
List all possible rational zeros using the
rational zero theorem.
f ( x )  4 x  x  3 x  9 x  10
4
3
2
List all possible rational zeros using the
rational zero theorem.
f ( x )  2 x  3 x  11 x  6
3
2
Find the zeros of a polynomial function.
• List the possible rational zeros of the function.
• Test the zeros using division. (Since the zeros
are x-intercepts, when you divide you should
end up with a remainder of zero.)
– Graph the function in the calculator to narrow your
list. Only check reasonable values from the list.
– The number of zeros is the same as the degree of
the polynomial.
Find all real zeros of the function.
f ( x )  x  8 x  11 x  20
3
2
Do Now: Find all real zeros of the function.
f ( x )  x  4 x  15 x  18
3
2
Academy
Algebra II
5.6: Find Rational Zeros
HW tonight: p.374 (16-26 even)
Quiz Friday: 5.5, 5.6
(Calculator and no calculator section)
Find all real zeros of the function.
f ( x )  10 x  11 x  42 x  7 x  12
4
3
2
Find all real zeros of the function.
f ( x )  48 x  4 x  20 x  3
3
2
Do Now: Find all real zeros of the function.
f ( x )  2 x  5 x  18 x  19 x  42
4
3
2