15 45 141 5 15 47 140 The remainder is 140, so = 140. f(3) Exercises Use synthetic substitution to find and for each function. f(–5) f 1. = + 5x 1 2. = 4 + 6x 7 f(x) f(x) –3 – – 3. = + 3 5 4. = + 11 1 f(x) f(x) – – – Use synthetic substitution to find and for each function. f(4) f(–3) 5. = 2 + 5x + 3 6. = 3 4x + 2 f(x) f(x) – – 7. = 5– 4 + 2 8. = 2 4 + 3 + 6 f(x) f(x) – x – 9. = 5 + 3 4 2x + 4 10. = 3 2 + 2x 5 f(x) f(x) – – – – – 5-6 Guide (continued) Factors of Polynomials The can help you find all the factors of a polynomial. Factor Theorem Factor Theorem The binomial x a is a factor of the polynomial f(x) if and only if f(a) = 0. – Example: Show that + 5 is a factor of + 2– 13x + 10. Then find the remaining factors of the polynomial. x By the Factor Theorem, the binomial + 5 is a factor of the polynomial if is a zero of the polynomial function. x –5 To check this, use synthetic substitution. 1 2 10 –5 –13 15 –5 –10 1 2 0 –3 Since the remainder is 0, + 5 is a factor of the polynomial. The polynomial + 2 13x + 10 can be factored as x – (x + 5)( 3x + 2). The depressed polynomial 3x + 2 can be factored as (x 2)(x 1). – – – – So + 2 13x + 10 = (x + 5)(x 2)(x 1). – – – Exercises Given a polynomial and one of its factors, find the remaining factors of the polynomial. 1. + 10x + 8; 2 2. 4 11x + 30; + 3 x x – – – –
15 45 141 5 15 47 140 The remainder is 140, so f(3) = 140. Exercises Use synthetic substitution to find f(–5) and f for each function. 1. f(x) = –3 + 5x – 1 2. f(x) = 4 + 6x – 7 3. f(x) = – + 3 – 5 4. f(x) = + 11 – 1 Use synthetic substitution to find f(4) and f(–3) for each function. 5. f(x) = 2 + – 5x + 3 6. f(x) = 3 – 4x + 2 7. f(x) = 5– 4 + 2 8. f(x) = 2 – 4 + 3 + x – 6 9. f(x) = 5 + 3 – 4 – 2x + 4 10. f(x) = 3 – 2 – + 2x – 5 5-6 Guide (continued) Factors of Polynomials The Factor Theorem can help you find all the factors of a polynomial. Factor Theorem The binomial x – a is a factor of the polynomial f(x) if and only if f(a) = 0. Example: Show that x + 5 is a factor of + 2– 13x + 10. Then find the remaining factors of the polynomial. By the Factor Theorem, the binomial x + 5 is a factor of the polynomial if –5 is a zero of the polynomial function. To check this, use synthetic substitution. –5 1 2 –13 10 –5 15 –10 1 –3 2 0 Since the remainder is 0, x + 5 is a factor of the polynomial. The polynomial + 2 – 13x + 10 can be factored as (x + 5)( – 3x + 2). The depressed polynomial – 3x + 2 can be factored as (x – 2)(x – 1). So + 2 – 13x + 10 = (x + 5)(x – 2)(x – 1). Exercises Given a polynomial and one of its factors, find the remaining factors of the polynomial. 1. + – 10x + 8; x – 2 2. – 4 – 11x + 30; x + 3
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