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Learn to write polynomial functions of least degree with rational coefficients using given zeros. Examples and guided practice included.
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Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and 3 and 2 + 5 as zeros. Because the coefficients are rational and 2 + 5is a zero, 2–5 must also be a zero by the irrational conjugates theorem. Use the three zeros and the factor theorem to write f (x) as a product of three factors. EXAMPLE 3 Use zeros to write a polynomial function SOLUTION
f (x) = (x – 3) [ x – (2 + √ 5 ) ] [ x – (2 – √ 5 ) ] = (x – 3) [ (x – 2) – √ 5 ] [ (x – 2) +√ 5 ] EXAMPLE 3 Use zeros to write a polynomial function Write f (x) in factored form. Regroup terms. = (x – 3)[(x – 2)2 – 5] Multiply. = (x – 3)[(x2 – 4x + 4) – 5] Expand binomial. = (x – 3)(x2 – 4x – 1) Simplify. = x3 – 4x2 – x – 3x2 + 12x + 3 Multiply. Combine like terms. = x3 – 7x2 + 11x + 3
f(2 + √ 5 ) = (2 + √ 5 )3 – 7(2 + √ 5 )2 + 11( 2 + √ 5 ) + 3 = 38 + 17 √5– 63 – 28 √5+ 22 + 11√5+ 3 Sincef (2 + √5 ) = 0, by the irrational conjugates theorem f (2 –√ 5) = 0. EXAMPLE 3 Use zeros to write a polynomial function CHECK You can check this result by evaluating f (x) at each of its three zeros. f(3) = 33 – 7(3)2 + 11(3) + 3 = 27 – 63 + 33 + 3 = 0 = 0
GUIDED PRACTICE for Example 3 Write a polynomial functionf of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros. 5. – 1, 2, 4 Use the three zeros and the factor theorem to write f(x) as a product of three factors. SOLUTION f (x) = (x + 1) (x – 2) ( x – 4) Write f (x) in factored form. = (x + 1) (x2– 4x – 2x + 8) Multiply. = (x + 1) (x2– 6x + 8) Combine like terms. = x3 – 6x2 + 8x+ x2– 6x + 8 Multiply. = x3– 5x2 + 2x + 8 Combine like terms.
6. 4, 1 + √5 f (x) = (x – 4) [ x – (1 + √ 5 ) ] [ x – (1 – √ 5 ) ] Because the coefficients are rational and 1 + 5 is a zero, 1 – 5 must also be a zero by the irrational conjugates theorem. Use the three zeros and the factor theorem to write f(x) as a product of three factors = (x – 4) [ (x – 1) – √ 5 ] [ (x – 1) +√ 5 ] = (x – 4)[(x – 1)2 – ( 5)2] GUIDED PRACTICE for Example 3 SOLUTION Write f (x) in factored form. Regroup terms. Multiply. = (x – 4)[(x2 – 2x + 1) – 5] Expand binomial.
GUIDED PRACTICE for Example 3 = (x – 4)(x2 – 2x – 4) Simplify. = x3 – 2x2 – 4x – 4x2 + 8x + 16 Multiply. = x3 – 6x2 + 4x +16 Combine like terms.
√6 7. 2, 2i, 4 – Because the coefficients are rational and 2i is a zero, –2i must also be a zero by the complex conjugates theorem. 4 + 6 is also a zero by the irrational conjugate theorem. Use the five zeros and the factor theorem to write f(x) as a product of five factors. f (x) = (x–2) (x +2i)(x-2i)[(x –(4–√6 )][x –(4+√6) ] = (x – 2) [ (x2–(2i)2][x2–4)+√6][(x– 4) – √6 ] = (x – 2)[(x2+ 4)[(x– 4)2 – ( 6 )2] GUIDED PRACTICE for Example 3 SOLUTION Write f (x) in factored form. Regroup terms. Multiply. = (x – 2)(x2 + 4)(x2– 8x+16 – 6) Expand binomial.
GUIDED PRACTICE for Example 3 = (x – 2)(x2 + 4)(x2– 8x + 10) Simplify. = (x–2) (x4– 8x2 +10x2 +4x2 –3x +40) Multiply. = (x–2) (x4 – 8x3+14x2 –32x + 40) Combine like terms. = x5– 8x4 +14x3 –32x2 +40x– 2x4 +16x3 –28x2 + 64x – 80 Multiply. = x5–10x4 + 30x3 – 60x2 +10x– 80 Combine like terms.
GUIDED PRACTICE for Example 3 8. 3, 3 – i Because the coefficients are rational and 3 –i is a zero, 3 + i must also be a zero by the complex conjugates theorem. Use the three zeros and the factor theorem to write f(x) as a product of three factors SOLUTION = f(x) =(x – 3)[x – (3 – i)][x –(3 + i)] Write f (x) in factored form. = (x–3)[(x– 3)+i ][(x2 – 3) – i] Regroup terms. = (x–3)[(x– 3)2 –i2)] Multiply. = (x– 3)[(x – 3)+ i][(x–3) –i]
GUIDED PRACTICE for Example 3 = (x – 3)[(x – 3)2 – i2]=(x –3)(x2 – 6x + 9) Simplify. = (x–3)(x2 – 6x + 9) = x3–6x2 + 9x– 3x2 +18x – 27 Multiply. Combine like terms. = x3 – 9x2 + 27x –27