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4.1-4.3 Review
Like the 3.4-3.6 test – use this sheet AND your 4.1-4.2 review to study!!
Vocabulary
polynomial, complex zero/root, imaginary zero/root, real zero/root, discriminant, degree,
Fundamental Theorem of Algebra, Remainder Theorem, Factor Theorem, depressed polynomial
4.1 Polynomial Functions
 Complex numbers
o Notation
o Addition, subtraction, multiplication, solving with i
 Fundamental Theorem of Algebra
o Use to find roots
o Use to write polynomials for given roots
4.2 Quadratic Equations
 Quadratic formula
 Completing the square
 Use the discriminant to tell the nature of a polynomial’s roots
o Three cases:
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4.3 The Remainder and Factor Theorems

The Remainder Theorem
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The Factor Theorem
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Synthetic division
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Finding the completely factored form of polynomials using synthetic division
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Finding values that make polynomials divisible by a given factor
APC1: 4.1-4.3 Review
Practice
Use a separate sheet of paper for your work
1. Write the polynomial equation of least degree with roots 3 and 4.
2. Write the polynomial of least degree with roots –2, i, and –i.
3. Write the polynomial of least degree with roots 2, –2, 3i, and –3i.
4. State the number of complex roots of x 3  11x 2  30 x  0 . Then find the roots.
5. Solve x2 – 4x – 5 by completing the square, the quadratic formula, and factoring.
6. Solve x 2  4 x  6  0 by completing the square and the quadratic formula.
7. Find the discriminant and describe the nature of the roots of x2 + 6x – 13 = 0. Then find
the roots using a method of your choosing.
8. Find the discriminant and describe the nature of the roots of 4n2 – 4n + 1 = 0. Then find
the roots using a method of your choosing.
9. Divide x3  3x 2  2 x  8 by x + 2 using synthetic division.
10. Divide ( x3  3x  5)  ( x  1) using synthetic division.
11. Use the Remainder Theorem to find the remainder for ( x3  4 x 2  2 x  6)  ( x  4) . State
whether the binomial is a factor of the polynomial.
12. Determine the completely factored form of x3  2 x 2  5 x  6 given that x = 1 is a root.
13. Determine the completely factored form of x3 – 7x + 6 given that x – 1 is a factor.
Addition problems can be found on page A32 of your book, the answers to which will be posted
to our website.
APC1: 4.1-4.3 Review