 # 5.7 Apply the Fundamental Theorem of Algebra

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5.7 Apply the Fundamental Theorem of Algebra

Fundamental Theorem of Algebra
The degree of the problem identifies the number of solutions to the problem. (Ex) How solutions are there to the problem below? x5 + 2x3 – 5x – 12

Write a Polynomial using Zero’s
(Ex) -1, 2, 4 Step 1: Determine the factors that created the zero’s. (x + 1) (x – 2) (x – 4) Step 2: Use Foil Method to simplify the polynomial.

Complex Conjugate Theorem
If a + bi is a zero, then a – bi is also a zero. If a + √b is a zero, then a - √b is also a zero.

Write a polynomial using the zero’s
(Ex) 3, 2 + √5 Step 1: Write the factors: (x – 3) (x – (2 + √5)) ( x – (2 - √5)) Step 2: Regroup the conjugates: Step 3: Foil Method

(Ex) Write a polynomial using the zero’s
4, 1+√5 2, 2i, 4-√6 3, 3 - i

HW Problems Write a polynomial using the zero’s. 5, -3, 1 2, 2+√3

Discarte’s Rule of Signs
The number of positive real zeros of f is equal to the number of the coefficients of f(x) or is less than this by an even number. The number of negative real zeros of f is equal to the number of the coefficients of f(-x) or is less than this by an even number.

Determine the possible number of positive real zeros, negative real zeros, and imaginary zeros.
(Ex) x6 – 2x5 + 3x4 – 10x3 – 6x2 + 8x – 8 Step 1: Determine the number of sign changes. Step 2: Determine how many solutions. Step 3: List the possibilities.

Determine the possible amount of positive real zeros, negative real zeros, and imaginary zeros.
(Ex) f(x) = x3 + 2x – 11 (Ex) g(x) = 2x4 – 8x3 + 6x2 – 3x + 1