Algebra II
Unit #4: Solving Polynomial Functions, Modeling
(5.4 (continued), 5.5, 5.6, 5.8)
Name: ________________________
Block: __________
Launch Activity: For questions 1-4 use your graphing calculator to complete all parts:
a) State the degree of the polynomial
b) Find the zeros of the polynomial using your calculator.
c) State the least possible multiplicity of each zero by looking at the graph.
d) Find the total number of zeros (count the multiplicities of all the zeros)
e) Sketch a graph of the function (be sure to label all zeros)
1. 𝑓(𝑥) = 𝑥 4 + 6𝑥 3 + 8𝑥 2 − 6𝑥 − 9
2. 𝑔(𝑥) = −𝑥 5 − 2𝑥 4 + 3𝑥 3 + 4𝑥 2 − 4𝑥
a)
a)
b)
b)
c)
c)
d)
d)
3. ℎ(𝑥) = 2𝑥 3 − 9𝑥 2 − 47𝑥 + 84
(Use: Y-min: -200, Y-max: 200)
4. 𝑚(𝑥) = 6𝑥 4 − 22𝑥 3 − 4𝑥 2 + 48𝑥
(Use: Y-min: -30, Y-max: 30)
a)
a)
b)
b)
c)
c)
d)
d)
5) What do you notice about the degree of the polynomial and the total number of zeros?
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This is called the Fundamental Theorem of Algebra: It says that every polynomial function of
degree n, (n ≥1) has _________________ ______ roots/zeros, including ________________
roots/zeros and _________________ (_____________________) roots/zeros.
In my own words:
**The real zeros will look like ______________ on the graph, you will not see the complex zeros!
Synthetic Division – Finding all Roots/Zeros and Factors for Polynomials
**The algorithm is the same as Synthetic Substitution, but we will take this one step further:
Steps for Using Synthetic Division to Solve a Polynomial Equation
1. Set up the synthetic division by listing the coefficients of the polynomial function in standard form.
**Use zero as the coefficient of any “missing” term!**
2. Find the synthetic divisor: (this is the x-intercept/zero/root of the function)
3. Bring down the first coefficient, multiply the coefficient by the divisor, add the column, repeat.
4. The last number in your quotient gives the REMAINDER of the division problem.
**A remainder of ____ means the divisor is a ZERO/ROOT/X-INTERCEPT of the polynomial function
(NEW!) 5. The quotient is a new polynomial function with the coefficients of your final line. All degrees
are one less than the original.
(NEW!) 6. Factor the quotient again if possible or repeat synthetic division with a new divisor.
*our goal is to get to a quadratic expression, and then we can factor or use the quadratic formula to solve!
Practice: Divide using synthetic division. Write the quotient as a polynomial function and list the
remainder. Show your work!
Example A: (x2 + 3) ÷ (x  1)
Example B: (x5 + 7x2 + 15x + 9) ÷ (x + 1)
Quotient: _____________________R: _____
Was this a factor? Yes / No
Quotient: _____________________R: _____
Was this a factor? Yes / No
#1) (x3 – 4x2 + 6x – 4) ÷ (x – 2)
#2) (x4  x3 + x2  x + 1) ÷ (x  1)
Quotient: _____________________R: _____
Was this a factor? Yes / No
Quotient: _____________________R: _____
Was this a factor? Yes / No
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#3) (2x4 + 7x3  11x2 + 21x + 5) ÷ (x + 5)
#4) (2x4 + 23x3 + 60x2  125x  500) ÷ (x + 4)
Quotient: _____________________R: _____
Was this a factor? Yes / No
Quotient: _____________________R: _____
Was this a factor? Yes / No
Practice: Using Synthetic Division to find the COMPLETE Factorization
Use synthetic division and the given factor to reduce to a quadratic expression. Then factor the
expression.
**Once it’s quadratic, we can factor without Synthetic Division. Then list ALL factors!
Example C: f(x) = x3 + 3x2  13x  15; (x + 5)
Example D: f(x)= x3  3x2  10x + 24; (x  2)
Factored Form: ___________________
Factored Form: ___________________
#5) f(x) = x3 + x2  10x + 8; (x  1)
#6) f(x) = x3 + 4x2  9x  36; (x + 3)
Factored Form: ___________________
Factored Form: ___________________
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Synthetic Division: Finding Multiplicities of Zeros
Use your graphing calculator to identify the real, rational zeros. Then use synthetic division to
identify the multiplicities of each zero. How can you recognize a multiple zero on a graph?
How can you determine the multiplicity of a zero/root using synthetic division? (hint: if something
is a repeated zero, what should happen with synthetic division?)
**The constant term (c value) of the polynomial function helps us set the WINDOW: the X-min
and X-max will be at most |𝑐| units from 0. This means:
Example E: 𝑓(𝑥) = 𝑥 3 − 9𝑥 2 + 24𝑥 − 16
(zeros: x = ___________________)
Synthetic Division: Finding Complex/ Imaginary Zeros
Example F: : 𝑓(𝑥) = 𝑥 4 + 6𝑥 3 + 10𝑥 2 + 6𝑥 + 9
How can Synthetic
Division help you find
the complex / imaginary
roots?
What real zeros can you determine with your calculator? ______________
(include any multiplicities so we divide this the right amount of times!)
Zeros: _______________________________
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Finding All Zeros/Roots of Polynomial Functions (here’s what you need to be able to do!)
a) Graph the function to find the real roots/zeros on your calculator. (these are the __ - ________________)
b) Use Synthetic Division to find all zeros (real and imaginary) of the function. Identify any multiple
zeros. **Get it to a quadratic or linear, then we can use factoring/quad. formula/square roots to find complex zeros!
c) List the factored form of the polynomial function using your work with Synthetic Division.
--Synthetic Division tests zeros; use these to list out factors ( )( )( ), use exponents for multiple zeros
Example G: 𝑓(𝑥) = 𝑥 4 + 2𝑥 3 + 5𝑥 2 + 8𝑥 + 4
a) x = ___________________
b)
Example H: 𝑓(𝑥) = 𝑥 3 – 11𝑥 + 6
a) x = ___________________
b)
c) f(x) = ___________________________
c) f(x) = ___________________________
#7) 𝑓(𝑥) = 𝑥 4 – 3𝑥 2  4
#8) 𝑓(𝑥) = 𝑥 3 − 4𝑥 2 + 9𝑥 − 36
a) x = ___________________
b)
a) x = ___________________
b)
c) f(x) = _________________________
c) f(x) = _________________________
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#9) 𝑓(𝑥) = 𝑥 4 + 𝑥 3 − 7𝑥 2 − 9𝑥 − 18
*#10) 𝑓(𝑥) = 2𝑥 4 − 11𝑥 3 − 21𝑥 2
a) x = ___________________
b)
a) x = ___________________
b)
c) f(x) = _________________________
c) f(x) = _________________________
#11) 𝑓(𝑥) = 𝑥 4 − 2𝑥 3 + 𝑥 2 − 2𝑥
#12) 𝑓(𝑥) = 𝑥 4 + 4𝑥 3 − 𝑥 2 − 20𝑥 − 20
a) x = ___________________
b)
a) x = ___________________
b)
c) f(x) = _________________________
c) f(x) = _________________________
**Do MathXL 5.4 – Synthetic Division and Substitution**
**Do MathXL 5.5-5.6 Homework**
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