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**Find real zeros of polynomial functions. **

You learned that a polynomial function of degree n can have at most n real zeros. (Lesson 2-1) Find real zeros of polynomial functions. Find complex zeros of polynomial functions. Then/Now

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Key Concept 1

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**Step 1 Identify possible rational zeros. **

Leading Coefficient Equal to 1 A. List all possible rational zeros of f (x) = x3 – 3x2 – 2x + 4. Then determine which, if any, are zeros. Step 1 Identify possible rational zeros. Because the leading coefficient is 1, the possible rational zeros are the integer factors of the constant term 4. Therefore, the possible rational zeros are ±1, ± 2, and ± 4. Example 1

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**Step 2 Use direct substitution to test each possible zero. **

Leading Coefficient Equal to 1 Step 2 Use direct substitution to test each possible zero. f(1) = (1)3 – 3(1)2 – 2(1) + 4 or 0 f(–1) = (–1)3 – 3(–1)2 – 2(–1) + 4 or 2 f(2) = (2)3 – 3(2)2 – 2(2) + 4 or –4 f(–2) = (–2)3 – 3(–2)2 – 2(–2) + 4 or –12 f(4) = (4)3 – 3(4)2 – 2(4) + 4 or 12 f(–4) = (–4)3 – 3(–4)2 – 2(–4) + 4 or –100 The only rational zero is 1. Answer: ±1, ± 2, ± 4; 1 Example 1

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**Leading Coefficient Equal to 1**

B. List all possible rational zeros of f (x) = x 3 – 2x – 1. Then determine which, if any, are zeros. Step 1 Because the leading coefficient is 1, the possible rational zeros are the integer factors of the constant term –1. Therefore, the possible rational zeros of f are 1 and –1. Example 1

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**Step 2 Begin by testing 1 and –1 using synthetic substitution.**

Leading Coefficient Equal to 1 Step 2 Begin by testing 1 and –1 using synthetic substitution. –2 –1 1 1 –1 1 1 –1 –2 –1 1 0 –2 –1 –1 1 1 1 –1 –1 0 Example 1

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**Leading Coefficient Equal to 1**

Because f (–1) = 0, you can conclude that x = –1 is a zero of f. Thus f (x) = (x + 1)(x 2 – x – 1). Because the factor x 2 – x – 1 yields no rational zeros, we can conclude that f has only one rational zero, x = –1. Answer: ±1; 1 Example 1

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**List all possible rational zeros of f (x) = x 4 – 12x 2 – 15x – 4**

List all possible rational zeros of f (x) = x 4 – 12x 2 – 15x – 4. Then determine which, if any, are zeros. A. B. C. D. Example 1

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**Leading Coefficient not Equal to 1**

List all possible rational zeros of f (x) = 2x 3 – 5x 2 – 28x Then determine which, if any, are zeros. Step 1 The leading coefficient is 2 and the constant term is 15. Possible rational zeros: Example 2

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**Leading Coefficient not Equal to 1**

Step 2 By synthetic substitution, you can determine that x = –3 is a rational zero. Testing each subsequent possible zero on the depressed polynomial, you can determine that x = 5 and are also rational zeros. By the division algorithm, f (x) = (x + 3)(x – 5)(2x – 1) so the rational zeros are x = –3, x = 5, and Check this result by graphing. Example 2

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**Leading Coefficient not Equal to 1**

Answer: Example 2

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**List all possible rational zeros of f (x) = 4x 3 – 20x 2 + x – 5**

List all possible rational zeros of f (x) = 4x 3 – 20x 2 + x – 5. Then determine which, if any, are zeros. A. B. C. D. Example 2

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Key Concept 2

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Determine an interval in which all real zeros of f (x) = 2x 4 – 5x 3 – 13x x – 10 must lie. Then find all the real zeros. A. [0, 4]; 1, 2 B. [–1, 2]; 1, C. [–3, 5]; 1, D. [–2, 1]; 1, Example 4

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Key Concept 3

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**Examine the variations of sign for f (x) and for f(–x).**

Use Descartes’ Rule of Signs Describe the possible real zeros of f (x) = x 4 – 3x 3 – 5x 2 + 2x + 7. Examine the variations of sign for f (x) and for f(–x). f (x) = x4 – 3x3 – 5x2 + 2x + 7 + to – – to + f(–x) = (–x)4 – 3(–x)3 – 5(–x)2 + 2(–x) + 7 + to – – to + = x 4 + 3x 3 – 5x 2 – 2x + 7 Example 5

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**Answer: 2 or 0 positive real zeros, 2 or 0 negative real zeros**

Use Descartes’ Rule of Signs The original function f (x) has two variations in sign, while f (–x) also has two variations in sign. By Descartes' Rule of Signs, you know that f (x) has either 2 or 0 positive real zeros and either 2 or 0 negative real zeros. Answer: 2 or 0 positive real zeros, 2 or 0 negative real zeros Example 5

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**Describe the possible real zeros of g (x) = –x 3 + 8x 2 – 7x + 9.**

A. 3 or 1 positive real zeros, 1 negative real zero B. 3 or 1 positive real zeros, 0 negative real zeros C. 2 or 0 positive real zeros, 0 negative real zeros D. 2 or 0 positive real zeros, 1 negative real zero Example 5

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Key Concept 4

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Key Concept 5

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**f (x) = a[x – (–1)](x – 2)[x – (2 – i)](x – (2 + i)]**

Find a Polynomial Function Given Its Zeros Write a polynomial function of least degree with real coefficients in standard form that has –1, 2, and 2 – i as zeros. Because 2 – i is a zero and the polynomial is to have real coefficients, you know that 2 + i must also be a zero. Using the Linear Factorization Theorem and the zeros –1, 2, 2 – i, and 2 + i, you can write f (x) as follows: f (x) = a[x – (–1)](x – 2)[x – (2 – i)](x – (2 + i)] Example 6

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**f (x) = (1)(x + 1)(x – 2)[x – (2 – i)][x – (2 + i)] Let a = 1. **

Find a Polynomial Function Given Its Zeros While a can be any nonzero real number, it is simplest to let a = 1. Then write the function in standard form. f (x) = (1)(x + 1)(x – 2)[x – (2 – i)][x – (2 + i)] Let a = 1. = (x2 – x – 2)(x2 – 4x + 5) Multiply. = x4 – 5x3 + 7x2 + 3x – 10 Multiply. Therefore, a function of least degree that has –1, 2, and 2 – i as zeros is f (x) = x4 – 5x3 + 7x2 + 3x – 10 or any nonzero multiple of f (x). Answer: Sample answer: f (x) = x4 – 5x3 + 7x2 + 3x – 10 Example 6

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Write a polynomial function of least degree with real coefficients in standard form that has –2 (multiplicity 2), 0, and 3i as zeros. A. f (x) = x 5 + 4x x x x B. f (x) = x 5 + 4x 4 + 9x x C. f (x) = x 3 + 2x 2 – 3ix 2 – 6xi D. f (x) = x 4 + 4x 3 – 5x 2 – 36x – 36 Example 6

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Key Concept 6

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**k (–x) = (–x)5 + (–x)4 – 13(–x)3 – 23(–x)2 – 14(–x) – 24 **

Factor and Find the Zeros of a Polynomial Function A. Consider k (x) = x5 + x4 – 13x3 – 23x2 – 14x – 24. Write k (x) as the product of linear and irreducible quadratic factors. The possible rational zeros are ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24. The original polynomial has 1 sign variation. k (–x) = (–x)5 + (–x)4 – 13(–x)3 – 23(–x)2 – 14(–x) – 24 = –x 5 + x x 3 – 23x x – 24 k(–x) has 4 sign variations, so k (x) has 1 positive real zero and 4, 2, or 0 negative real zeros. Example 7

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**Factor and Find the Zeros of a Polynomial Function**

The graph shown suggests 4 as one real zero of k (x). Use synthetic substitution to test this possibility. Example 7

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**Factor and Find the Zeros of a Polynomial Function**

4 1 1 –13 –23 –14 –24 Because k (x) has only 1 positive real zero, you do not need to test any other possible positive rational zeros. The graph suggests that –2 and –3 are negative real zeros. Test these possibilities successively in the depressed quartic and then cubic polynomials. Example 7

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**–3 1 3 1 3 Now test –3 in the depressed polynomial. –3 0 –3 1 0 1 0**

Factor and Find the Zeros of a Polynomial Function – Begin by testing –2. –2 –6 –2 –6 – Now test –3 in the depressed polynomial. –3 0 –3 The remaining quadratic factor (x 2 + 1) yields no real zeros and is therefore irreducible over the reals. So, k (x) written as a product of linear and irreducible quadratic factors is k (x) = (x – 4)(x + 2)(x + 3)(x 2 + 1). Example 7

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**Answer: k (x) = (x – 4)(x + 2)(x + 3)(x2 + 1)**

Factor and Find the Zeros of a Polynomial Function Answer: k (x) = (x – 4)(x + 2)(x + 3)(x2 + 1) Example 7

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**Answer: k (x) = (x – 4)(x + 2)(x + 3)(x + i)(x – i)**

Factor and Find the Zeros of a Polynomial Function B. Consider k (x) = x5 + x4 – 13x3 – 23x2 – 14x – 24. Write k (x) as the product of linear factors. You can factor x by writing the expression first as a difference of squares or x 2 – i 2. Then factor this difference of squares as (x – i)(x + i). So, k (x) written as a product of linear factors is k (x) = (x – 4)(x + 2)(x + 3)(x + i)(x – i). Answer: k (x) = (x – 4)(x + 2)(x + 3)(x + i)(x – i) Example 7

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**Factor and Find the Zeros of a Polynomial Function**

C. Consider k (x) = x 5 + x 4 – 13x 3 – 23x 2 – 14x – 24. List all the zeros of k (x). Because the function has degree 5, by the corollary to the Fundamental Theorem of Algebra k (x) has exactly five zeros, including any that may be repeated. The linear factorization gives us these five zeros: 4, –2, –3, –i, and i. Answer: 4, –2, –3, –i, i Example 7

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**A. (x + 1)(x – 1)(x + (2 – i))(x + (2 + i)) **

Write k (x) = x 4 – 4x 3 + 4x 2 + 4x – 5 as the product of linear factors. A. (x + 1)(x – 1)(x + (2 – i))(x + (2 + i)) B. (x + 1)(x – 1)(x 2 – 4x + 5) C. (x + 1)(x – 1)(x – (2 + i))(x – (2 – i)) D. (x + 1)(x – 1)(x + 5) Example 7

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**Use synthetic substitution to verify that 2 + 5i is a zero of p (x).**

Find the Zeros of a Polynomial When One is Known Find all complex zeros of p (x) = x 4 – 6x x 2 – 50x – 58 given that x = 2 + 5i is a zero of p. Then write the linear factorization of p (x). Use synthetic substitution to verify that 2 + 5i is a zero of p (x). 2 + 5i 1 –6 35 –50 –58 (2 + 5i)(–4 + 5i) = –8 – 10i + 25i 2 2 + 5i –33 – 10i = –8 – 10i + 25(–1) 1 –4 + 5i = –33 – 10i Example 8

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**Find the Zeros of a Polynomial When One is Known**

2 + 5i 1 –6 35 –50 –58 (2 + 5i)(2 – 10i) = 4 – 10i – 50i2 2 + 5i –33 – 10i 54 – 10i = 4 – 10i – 50(–1) 1 –4 + 5i 2 – 10i = 54 – 10i 2 + 5i 1 –6 35 –50 –58 (2 + 5i)(4 – 10i) = 8 – 50i 2 2 + 5i –33 – 10i 54 – 10i 58 = 8 – 50(–1) 1 –4 + 5i 2 – 10i 4 – 10i 0 = 58 Example 8

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**Find the Zeros of a Polynomial When One is Known**

Because x = 2 + 5i is a zero of p, you know that x = 2 – 5i is also a zero of p. Divide the depressed polynomial by 2 – 5i. 2 – 5i 1 –4 + 5i 2 – 10i 4 – 10i 2 – 5i –4 + 10i –4 + 10i 1 –2 –2 0 Using these two zeros and the depressed polynomial from this last division, you can write p (x) = [x – (2 + 5i)][x – (2 – 5i)](x 2 – 2x – 2). Example 8

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**Find the Zeros of a Polynomial When One is Known**

Because p (x) is a quartic polynomial, you know that it has exactly 4 zeros. Having found 2, you know that 2 more remain. Solve the remaining depressed polynomial, x 2 – 2x – 2, using the Quadratic Formula. Quadratic Formula a = 1, b = –2, and c = –2 Simplify. Simplify. Example 8

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**Find the Zeros of a Polynomial When One is Known**

Therefore, the four zeros of p are 2 + 5i, 2 – 5i, The linear factorization of p is p (x) = [x – (2 + 5i)](x – (2 – 5i)][x – ( )][x – ( )]. Using the graph of p, you can verify that the function has two real zeros at or 2.73 and at or –0.73. Example 8

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**Find the Zeros of a Polynomial When One is Known**

Answer: 2 + 5i, 2 – 5i, , ; p(x) = [x – (2 + 5i)][x – (2 – 5i)][x – ( )] ● [x – (1 – )] Example 8

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**Find all complex zeros of h(x) = x 4 + x 3 – 3x 2 + 9x – 108 given that x = –3i is a zero of h.**

A. 3i, –3i B. 3i, 4, –3 C. 3i, –3i, 4, –3 D. 3i, –3i, –4, 3 Example 8

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End of the Lesson

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3.3 - 1 3.3 Zero of Polynomial Functions Factor Theorem Rational Zeros Theorem Number of Zeros Conjugate Zeros Theorem Finding Zeros of a Polynomial Function.

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