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Pre-Calculus Section 2.3 Synthetic Division
Pre Calc Sec 2-3 Pre-Calculus Section 2.3 Synthetic Division
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Synthetic Division Use when dividing by linear binomial (x - k) polynomial must be in order put in zeros for missing terms multiply then add write answer with variables in it
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Use synthetic division to divide. 1. - 10x2 - 2x + x4 + 4 by x + 3
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Remainder Theorem If a polynomial f(x) is divided by x - k, the remainder is r = f(k) This means that when we use synthetic division the remainder is equal to the value of the divisor put into the function.
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Use the Remainder Theorem to evaluate the following function at x = - 2 2. f(x) = 3x3 + 8x2 + 5x - 7
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Factor Theorem A polynomial f(x) has a factor (x - k) if and only if f(k) = 0, which means that the remainder = 0 Means that when we use synthetic division if the remainder is equal to 0 we have found a factor / zero / solution of the function.
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Show that (x - 2) and (x + 3) are factors of 3
Show that (x - 2) and (x + 3) are factors of 3. f(x) = 2x4 + 7x3 - 4x2 - 27x - 18
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Use synthetic division to show that x is a solution, factor completely, and list all the real zeros of the function. 4. f(x) = x3 - 7x + 6 = 0 x = 2
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5. f(x) = 2x3 - 15x2 + 27x - 10 = x = ½
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Verify the given factors of the function f
Verify the given factors of the function f. Find the remaining factors of f, write the complete factorization of f, and list all real zeros of f. 6. f(x) = x4 - 4x3 - 15x2 + 58x - 40 factors: (x - 5) (x + 4)
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Steps to Find Rational Zeros
Rational Zero Test Used to find the zeros of polynomials that are of a degree of 3 or higher. Steps to Find Rational Zeros 1st - Use the graphing calculator – list the zeros 2nd - Use synthetic division to prove all of the zeros 3rd – Write as factors if directions state
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7. Find the rational zeros of f(x) = 2x3 + 3x2 - 8x + 3
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8. Find all the real zeros of f(x) = 10x3 - 15x2 - 16x + 12
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9. Find all the real zeros of f(x) = x4 - x3 - 2x - 4
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10. Find all the real zeros of f(x) = 4x5 + 12x4 - 11x3 - 42x2 + 7x + 30
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Verify the given factors of the function f and find the remaining factors of f. Then write the complete factorization of f. 11. 𝑓 𝑥 =2𝑥 3 + 𝑥 2 −5𝑥 + 2 𝑓𝑎𝑐𝑡𝑜𝑟 (𝑥+2)
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Verify the given factors of the function f and find the remaining factors of f. Then write the complete factorization of f. 12. 𝑓 𝑥 = 𝑥 4 − 4𝑥 3 − 15 𝑥 2 +58𝑥 − 40 𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑥+4 , (𝑥 − 5)
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