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**5.7 Apply the Fundamental Theorem of Algebra**

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**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

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**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.

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**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.

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**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

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**(Ex) Write a polynomial using the zero’s**

4, 1+√5 2, 2i, 4-√6 3, 3 - i

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

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**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.

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**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.

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**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

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