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20 Cards in this Set
- Front
- Back
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If y = P(x) is a polynomial function, then the zero of the polynomial function is:
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the value of x that satisfies P(x) = 0.
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The zeros of P(x) = x² - 9 are:
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-3 and 3.
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The zeros of P(x) = x² - 9 are the same as the solutions to the equation:
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x² - 9 = 0
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The real zeros of a polynomial function appear on the graph of the function as:
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the x-coordinates of the x-intercepts.
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zeros can be found by solving linear or quadratic equations for polynomial functions with a degree of:
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2 or less
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What is the remainder theorem?
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P(c) = R when P(x) is divided by x - c.
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What are the steps for synthetic division?
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1) List all the coefficients of the polynomial and Place c as the divisor.
2) Bring down the first coefficient under the line. 3) Multiply c by this first coefficient brought down under the line. 4) Place this result under the next coefficient and add them. 5) Place the sum under the two addends. 6) Continue... the last value is the remainder. |
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The degree of the quotient in synthetic division will always be:
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one less than the degree of the dividend.
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What is the factor theorem?
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x - c is a factor of the polynomial P(x) IFF c is a zero of the P(x).
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p and q In the Rational Zero Theorem represent:
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p is a factor of the constant term.
q is a factor of the leading coefficient. |
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Describe the rational zero theorem:
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The rational number p/q is a zero of f(x).
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What is the remainder theorem used for?
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To evaluate polynomials
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What are the steps to using the remainder theorem to evaluate a polynomial? For example..
P(3) = 2x³ - 5x² + 4x - 6 |
1) Use synthetic division with c (in this case 3) as the divisor.
2) The remainder will be the solution to function P(x). |
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What is the factor theorem used for?
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To determine if a given expression is a factor of a polynomial.
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What are the steps to determine if an expression is the factor of a polynomial using the factor theorem?
eg.. if x + 4 is a factor of P(x) = x³ -13x + 12 |
x + 4 is a factor of P(x) IFF P(-4) = 0
1) Change sign of c. 2) Use synthetic division with c as the divisor. 3) The expression is a factor IFF the remainder is 0. 4) The quotient is an additional factor of the function. |
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What is the rational zero theorem used for?
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To find all possible rational zeros to a polynomial function.
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What are the steps for finding all of the rational zeros using the rational zero theorem?
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1) Find all of the factors of p (the constant term).
2) Find all of the factors of q (the leading coefficient). 3) Divide each factor of p by each factor of q. 4) Eliminate duplicates and add in negative factors. 5) Check each zero (including negatives) by using synthetic division. (R should = 0.) |
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When do you change the sign of c and when do you leave it alone in synthetic division?
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-Change the sign of c when doing synthetic division in the factor theorem.
-Leave it alone when using synthetic division to evaluate a polynomial. |
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When using synthetic division and the factor theorem, what is the proof that a given factor IS a factor of the function?
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c must be a zero of the function (the remainder is 0).
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How can you find the other factors if you prove a binomial is a factor of the function?
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The other factor is the quotient from the synthetic division.
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