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Warmup #1. Find the zeros, multiplicity and yintercept of f(x)= (x5)3(x+1)2(x7). Then, graph the function.
#2. Find a polynomial that has zeros: 3, 3,4,0
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2.3 Real zeros of functions To find real zeros: 1. long division 2. synthetic division 3. factor/remainder theorem 4. Test ±(p/q)
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Long division: must be used when divisor's deg. exceeds one Ex. Divide (2x4 + 4x3 5x2 + 3x – 2) by (x2 + 2x 3)
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ex. Divide (6x3 + 10x2 + x + 8) by (2x2 + 1)
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Rational zero test: must be used when asked to find zeros/ roots/solutions/xintercepts and traditional factoring techniques fail and/or the quadratic formula cannot be employed
Possible Rational Zeros: a list of "possible" zeros for the given function; not guaranteed to work; not a "complete" list as it doesn't contain irrational zeros or imaginary zeros. For a function f(x) with integer coefficients, Possible Rational Zeros can be written in the form :
, where p are the factors of the constant term and q are the factors of the leading coefficient
ex. List the possible rational zeros. f(x) = 2x3 + 3x2 8x + 3
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Notes 2.3 New COMPLETED.notebook
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Synthetic division • Works only when the divisor is a linear polynomial • (xk) is the generic representation for a linear factor • Used to find a) zeros and b) to write the factored form of a polynomial Ex. Find all real zeros. f(x) = 2x3 + 3x2 8x + 3
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Find all real zeros. Ex. f(x) = 2x4 + 7x3 4x2 27x 18
Ex3. f(x) = x3 + x + 2
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Notes 2.3 New COMPLETED.notebook
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Find all the real zeros of f(x)=10x3 15x2 16x +12 (When there are many possible zeros, use a calculator to help!)
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Remainder Theorem • If the polynomial f(x) is divided by x c, then the remainder is f(c). • Ex. Given use the Remainder Theorem to find f(2). Verify your answer using synthetic division. on.
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Ex. Use synthetic division to show that x=2 is a solution of
x3 + 2x2 2x 4=0. Use the result to factor the polynomial completely. List all real zeros.
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Notes 2.3 New COMPLETED.notebook
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Factor theorem • x − r is a factor of a polynomial P(x) if and only if r is a root of P(x). If r is a root, then when you use the remainder theorem, you should obtain zero. (or if you synthetically divide, the remainder should be zero. Ex. Are (x+4) and (x5) factors of f(x) = x³ − 3x² − 13x + 15?
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Notes 2.3 New COMPLETED.notebook
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Ex. Verify that (x+2) and (x1) are factors of
f(x)=2x3 +x2 5x +2. Find the remaining factors and zeros and write in factored form.
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Homework: p. 170 # 727 odd, 3549 odd, 5567odd
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