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Finding Zeros of Polynomial Functions:

The degree of a polynomial functions determines the number of zeros it will have (real or complex). When polynomial functions cross the x-axis, the x-value is called a real zero of the function. Some of these real zeros can be easily found by finding the x-intercepts of the graph (or by looking in the table).

However, not all real zeros are integers and can be easily found. To find all the zeros of a polynomial function, we’ll use synthetic division. In other words, if you know 3 is a factor of 6, then you can find the other factors by dividing 6 by 3. And as we’ve discussed previously, factors lead to zeros!

Read over the example below carefully!

|Function |Degree |# Of Zeros |Graph |# Of Real Zeros |List of All Zeros |

|Example[pic] |What is the highest|The degree determines |[pic] |How many times |See below how I found |

| |exponent? |the total number of | |does the graph |them. |

| | |zeros. | |cross the x-axis?| |

| | | | | |3, [pic] |

| |3 |3 | |1 | |

|Look in the table to identify the x-intercept: |Now use the x-value as the value for k in synthetic division to find the other |

|The x-value is a zero |zeros. Remember, a remainder of zero verifies that 3 is a real zero of the |

|so add it to the list. |function. |

| |[pic] |

| | |

| |The quotient is [pic]. Set the quotient equal to zero and solve. If you can’t |

| |factor the quadratic, then use square roots, the quadratic formula, or completing|

| |the square. |

| | |

|Note: check the table and the graph for all real zeros. You will need to use |[pic] I’ll solve this one using square roots. |

|synthetic division to find all zeros that are not integers or are imaginary. |The 2 remaining zeros are imaginary. Add them to the list above. Now you have |

| |found all three zeros! |

|In this case, only one of the real zeros is an integer. Therefore, we must use | |

|synthetic division to find the other two. | |

**Now you try. Complete the worksheet.

Finding Zeros of Polynomial Functions:

The degree of a polynomial functions determines the number of zeros it will have (real or complex). When polynomial functions cross the x-axis, the x-value is called a real zero of the function. Some of these real zeros can be easily found by finding the x-intercepts of the graph (or by looking in the table).

However, not all real zeros are integers and can be easily found. To find all the zeros of a polynomial function, we’ll use synthetic division. In other words, if you know 3 is a factor of 6, then you can find the other factors by dividing 6 by 3. And as we’ve discussed previously, factors lead to zeros!

Read over the example below carefully!

|Function |Degree |# Of Zeros |Graph |# Of Real Zeros |List of All Zeros |

|Example[pic] |What is the highest|The degree determines |[pic] |How many times |See below how I found |

| |exponent? |the total number of | |does the graph |them. |

| | |zeros. | |cross the x-axis?| |

| | | | | |3, [pic] |

| |3 |3 | |1 | |

|Look in the table to identify the x-intercept: |Now use the x-value as the value for k in synthetic division to find the other |

|The x-value is a zero |zeros. Remember, a remainder of zero verifies that 3 is a real zero of the |

|so add it to the list. |function. |

| |[pic] |

| | |

| |The quotient is [pic]. Set the quotient equal to zero and solve. If you can’t |

| |factor the quadratic, then use square roots, the quadratic formula, or completing|

| |the square. |

| | |

|Note: check the table and the graph for all real zeros. You will need to use |[pic] I’ll solve this one using square roots. |

|synthetic division to find all zeros that are not integers or are imaginary. |The 2 remaining zeros are imaginary. Add them to the list above. Now you have |

| |found all three zeros! |

|In this case, only one of the real zeros is an integer. Therefore, we must use | |

|synthetic division to find the other two. | |

**Now you try. Complete the worksheet

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