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Finding Real Zeros of Polynomial Functions

Learn how to find the real zeros of polynomial functions by using synthetic division and factoring. Practice with examples and guided exercises.

eugenesanders
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Finding Real Zeros of Polynomial Functions

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  1. List the possible rational zeros of f : 1 3 + , + , + , + , + , + , , + 6 12 + 1 , + 1 2 3 4 1 1 1 1 2 1 1 2 5 2 3 ,+ , +, + , + , +, + , + 3 4 12 1 6 5 5 5 5 5 10 10 Find zeros when the leading coefficient is not 1 EXAMPLE 3 Find all real zeros off (x) =10x4 – 11x3 – 42x2 + 7x + 12. SOLUTION STEP 1

  2. Choose reasonable values from the list above to check using the graph of the function. For f, the values 3 12 3 1 x = – , x = , x = , andx = 2 5 2 5 are reasonable based on the graph shown at the right. Find zeros when the leading coefficient is not 1 EXAMPLE 3 STEP 2

  3. 3 – 10 –11 –42 7 12 2 9 69 –15 39 – 4 2 21 23 10 – 26 – 3 – 4 2 –1 10 – 11 – 42 7 12 2 – 5 8 17 –12 10 – 16 – 34 24 0 ↑ 1 – is a zero. 2 Find zeros when the leading coefficient is not 1 EXAMPLE 3 STEP 3 Check the values using synthetic division until a zero is found.

  4. 1 f (x) = x + (10x3 – 16x2 – 34x + 24) 2 1 = x + (2)(5x3 – 8x2 – 17x + 12) 2 Find zeros when the leading coefficient is not 1 EXAMPLE 3 STEP 4 Factor out a binomial using the result of the synthetic division. Write as a product of factors. Factor 2 out of the second factor. = (2x +1)(5x3 – 8x2 – 17x +12) Multiply the first factor by2.

  5. Repeat the steps above for g(x) = 5x3 – 8x2 – 17x + 12. Any zero of g will also be a zero of f. The possible rational zeros of gare: 12 1 3 4 6 2 x = + 1, + 2, + 3, + 4, + 6, + 12, + , + , + , + , + , + 5 5 5 5 5 5 3 The graph of gshows that may be a zero. Synthetic division shows that is a zero and 5 3 5 3 g (x) = x – (5x2 – 5x – 20) 5 It follows that: f (x) = (2x + 1) g (x) Find zeros when the leading coefficient is not 1 EXAMPLE 3 STEP 5 = (5x – 3)(x2 – x – 4). = (2x + 1)(5x – 3)(x2 – x – 4)

  6. x = –(–1) +√ (–1)2 – 4(1)(–4) 2(1) x = 1 +√17 2 Find zeros when the leading coefficient is not 1 EXAMPLE 3 Find the remaining zeros of f by solving x2 – x – 4 = 0. STEP 6 Substitute 1 for a, –1for b, and –4 for cin the quadratic formula. Simplify. ANSWER 1 3 The real zeros of f are , , 1 + √17, and1 – √17. – 2 5 2 2

  7. ANSWER 2, , 1, +2 √2 3 1 3 1 2 4 6 2 ANSWER – , , – for Example 3 GUIDED PRACTICE Find all real zeros of the function. 5. f (x) = 48x3+ 4x2 – 20x + 3 6. f (x) = 2x4 + 5x3 – 18x2 – 19x + 42

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