# 008A Sample Final A, Question 9

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**Question: ** a) List all the possible rational zeros of the function

- b) Find all the zeros, that is, solve f(x) = 0

Foundations |
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1) What does the Rational Zeros Theorem say about possible zeros? |

2) How do you check if a possible zero is actually a zero? |

3) How do you find the rest of the zeros? |

Answer: |

1) The possible divisors can be found by finding the factors of -10, in a list, and the factors of 1, in a second list. Then write down all the fractions with numerators from the first list and denominators from the second list. |

2) Use synthetic division, or plug a possible zero into the function. If you get 0, you have found a zero. |

3) After your reduce the polynomial with synthetic division, try and find another zero from the list you made in part a). Once you reach a degree 2 polynomial you can finish the problem with the quadratic formula. |

Solution:

Step 1: |
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Start by factoring -10, and 1. Then the Rational Zeros Theorem gives us that the possible rational zeros are and . |

Step 2: |
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Start testing zeros with 1 and -1 since they require the least arithmetic. You will also find that 1 is a zero. Applying synthetic division you can reduce the polynomial to . |

Step 3: |
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Now we just need to find the zeros of . Since we are not down to a quadratic polynomial we have to find another zero from the list of rational zeros we found in step 1. You will find 2 is another root, and the polynomial can further be reduced to |

Step 4: |
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Now that the polynomial has been reduced to a quadratic polynomial we can use the quadratic formula to find the rest of the zeros. By doing so we find the roots are . Thus the zeros of are and |

Final Answer: |
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The roots are and |