Section 4 1A: The Rational Zeros (Roots) of a Polynomial The standard form for a general polynomial of degree n is written f (x) = a n x n + a n 1 x n 1 +... + a 1 x + a 0 where the highest degree term is a n x n and the lowest degree term is a 0 and a n x n 0 Examples of a polynomial in standard form f (x) = x 4 + x 3 17x + 6 f (x) = 4x 5 x 4 + x 7 f (x) = 3x 6 + x 4 4x 4 Rational Zeros of a Polynomial The rational number a is a zero of a function if f (a) = 0. For most second degree polynomials the common way to find the rational zeros for the polynomial in standard form is to factor the polynomial and set each factor equal to zero and solve for x. Examples of a finding the zeros of a second degree polynomial by factoring Example 1! Example! Example 3 x x 8 = 0 factor (x 4)(x + ) = 0 4 or x 9x + 4 = 0 factor (x 1)(x 4) = 0 1/ or 4 x 10x + 5 = 0 factor (x 5)(x 5) = 0 5 or 5 This section deals with finding only the rational zeros of a polynomial. How many rational zeros does a polynomial have? A polynomial of degree n is written f (x) = a n x n + a n 1 x n 1 +... + a 1 x + a 0. It has AT MOST n rational zeros. It may have less than n zeros. It many have none. Some of the rational zeros may be repeated but each occurrence of the same zero counts as a separate zero. (see example 3 above) Are there other zeros of a polynomial? If a polynomial cannot be factored using rational numbers then the zeros of the polynomial are NOT rational. The zeros may be irrational numbers like 4 5 or imaginary numbers like 3 + i. The quadratic formula can be used to find the irrational or imaginary zeros for second degree polynomials. Numerical methods such as Newton's method can also be used to find the irrational or imaginary zeros for higher powered polynomials. Section 4 1A Lecture Page 1 of 8 018 Eitel
A polynomial of degree n has n zeros. Some of the zeros may be irrational numbers. Irrational zeros come in conjugate pairs a + b c and a b c. Some zeros may be imaginary numbers. Imaginary zeros come in conjugate pairs a + b c and a b c. Some of the zeros may be rational numbers. They can be an even or odd number of rational zeros. A rational zero may be repeated. Each occurrence of the same zero counts as a separate zero. Finding the zeros of higher powered polynomials For most second degree polynomials the common way to find the rational zeros of the polynomial in standard form is to factor the polynomial and set each factor equal to zero and solve for x. For higher ordered polynomials factoring is too difficult in most cases. We will need to develop a new approach to finding the zeros. The approach we will use is based on what it means to be a factor or a zero of polynomial. Theorem If c is a zero of the polynomial f (x) = a n x n + a n 1 x n 1 +... + a 1 x + a 0 then (x c) is a factor of f (x) and (x c) must divide into f (x) with a zero remainder. This implies that synthetic division with c must produce a zero remainder Example f (x) = x 3x 10 can be factored into (x 5) and (x + ) so the zeros of f (x) are 5 and Prove this using the division test given above. Use synthetic division as the division process. Show 5 is a zero 5 1 3 10 5 10 1 0 (x 5) divides into f (x) with a zero remanider so (x 5) is a factor of f (x) and 5 is a zero of f (x) Show is a zero 1 3 10 10 1 5 0 (x + ) divides into f (x) with a zero remanider so (x + ) is a factor of f (x) and is a zero of f (x) We have seen that if we know c is a zero of f (x) the synthetic division with c must produce a zero remainder. How can this help us FIND the real zeros of a polynomial?. 1. List all the POSSIBLE zeros.. Use synthetic division to test each possible real zero until we have found all the real zeros. Section 4 1A Lecture Page of 8 018 Eitel
Finding all the Possible Rational Zeros of a Polynomial The Rational Root Theorem For any polynomial f (x) = a n x n + a n 1 x n 1 +... + a 1 x + a 0 The list of all possible roots or zeros of f (x) contains ± all the factors of a 0 all the factors of a n Example 1! Example The list of all possiable zeros of The list of all possiable zeros of f (x) = 5x 4 + x 3 17x + 4 f (x) = x 4 + x 3 17x 8 all the factors of 8 all the factors of 4 all the factors of 5 ± 1 1, ± 1, ± 4 1, ± 8 1, reduce the fractions and list in order ±1, ±, ± 4, ± 8 ± 1 1, ± 1, ± 4 1, ± 1 5, ± 5, ± 4 5 reduce the fractions and list in order ±1, ±, ± 4, ± 1 5, ± 5, ± 4 5 Example 3! Example 4 The list of all possiable zeros of The list of all possiable zeros of f (x) = 4x 5 + x 4 + x 3 17x 4 f (x) = x 4 + x 3 17x + 6 all the factors of 4 all the factors of 4 all the factors of 6 all the factors of ± 1 1, ± 1, ± 1 4, ± 1, ±, ± 4, ± 4 1, ± 4, ± 4 4 ± 1 1, ± 1, ± 1, ±, ± 3 1, ± 3, ± 6 1, ± 6 reduce the fractions eliminate the repeats and list the integers first reduce the fractions eliminate the repeats and list the integers first ±1, ±, ± 4, ± 1, ± 1 4 ±1, ± ± 3, ± 6, ± 1, ± 3 Section 4 1A Lecture Page 3 of 8 018 Eitel
Finding all the Rational Zeros for a Polynomial Example 1 Find all the possible rational zeros of f (x) = x 3 x x + possiable rational zeros all the factors of ± 1 1, ± 1, ± 1, ± ±1, ±, ± 1 test 1 as a root 1 1 1 1 1 1 1 0 1 is a zero and x x remains x x factors into (x )(x +1) so x + x has zeros of and 1 The three rational zeros of f (x) = x 3 x x + are 1, 1, and and the factors of f (x) = x 3 x x + are ( x + 1) (x 1) (x ) Section 4 1A Lecture Page 4 of 8 018 Eitel
Example Find all the rational zeros of f (x) = x 4 x 3 7x + 8x +1 possiable rational zeros ± 1 1, ± 1, ± 3 1, ± 4 1, ± 6 1, ± 1 1 ±1, ±, ± 3, ± 4, ± 6 Step 1! Step! Step A test 1 as a zero test 1 as a zero 1 1 7 8 1 1 1 7 8 1 1 1 8 0 1 3 4 1 1 1 8 0 (1) 1 3 4 1 (0) 1 is not a zero 1 is a zero and 1 3 4 1 remains we now use 1 3 4 1 1 to test for the remaining zeros Step 3! Step 4 test as a zero 1 3 4 1 1 1 1 6 (0) is a zero and x x 6 remains x x 6 factors into (x + )(x 3) so x x 6 has zeros of 3 and The four rational zeros of f (x) = x 4 x 3 7x + 8x +1 are 1,, and 3 Section 4 1A Lecture Page 5 of 8 018 Eitel
Example 3 Find all the rational roots of f (x) = 4x 4 +1x 3 +13x + 6x +1 possiable rational zeross all the factors of 4 ± 1 1, ± 1, ± 1 4 ±1, ± 1, ± 1 4 Step 1! Step! Step A test 1 as a root test 1 as a root 1 4 1 13 6 1 1 4 1 13 6 1 4 16 9 35 4 8 5 1 4 16 9 35 (36) 4 8 5 1 (0) 1 is not a zero 1 is a zero and 4 8 5 1 remains we now use 4 8 5 1 4 to test for the remaining zeros Step 3! Step 4! Step 5 test 1 as a zero test 1 as a zero 1 4 8 5 1 5 5 4 10 10 (6) 1 is not a zero 1 4 8 5 1 3 1 4 6 (0) 1 is a zero and 4x + 6x + remains 4x + 6x + factors into (x +1)(x +1) so 4x + 6x + has zeros of 1 and 1 The four rational zero of f (x) = 4x 4 +1x 3 +13x + 6x +1 are 1, 1, 1/ and 1/ Note: If you had retested 1 in step 3 you you have found it was a zero twice and had 4x + 4x +1 left over which factors into (x + 1) (x + 1) and not needed to test 1/ or 1/ Section 4 1A Lecture Page 6 of 8 018 Eitel
Example 4 Find all the rational roots of f (x) = x 3 + 5x 4x possiable rational zeros all the factors of 1 1, ± 1 ±1, ± Step 1! Step! Step 3 test 1 as a root 1 1 5 4 1 6 1 6 0 1 is a zero and x +6x + remains x + 6x + does not factor Use the quadratic equation 6 ± 36 4(1)() 6 ± 36 8 6 ± 8 6 ± 7 3 + 7, 3 7 The three zeros of f (x) = x 3 + 5x 4x are the rational zero 1 and the two irrational zeros 3 + 7, 3 7 Section 4 1A Lecture Page 7 of 8 018 Eitel
Example 5 Find all the rational roots of f (x) = x 3 5x + 9x +13 possiable rational zeros 3 1 1, ± 13 1 ±1, ±13 Step 1! Step! Step 3 test 1 as a root test 1 as a root 1 1 5 9 13 1 1 5 9 13 1 4 5 1 4 5 (18) 1 is not a zero 1 4 13 1 4 13 0 1 is a zero and x 4x +13 remains x 4x +13 does not factor Use the quadratic equation Step 4 4 ± 16 4(1)(13) 4 ± 16 5 4 ± 36 4 ± 6i + 3i, 3i The three zeros of f (x) = x 3 + 5x 4x are the rational zero 1 and the two complex zeros + 3i, 3i Section 4 1A Lecture Page 8 of 8 018 Eitel