SECTION 3.5SECTION 3.5

REAL ZEROS OF A POLYNOMIAL REAL ZEROS OF A POLYNOMIAL FUNCTIONFUNCTION

THE REAL ZEROS OF A POLYNOMIAL FUNCTIONTHE REAL ZEROS OF A

POLYNOMIAL FUNCTION

When we divide one polynomial When we divide one polynomial by another, we obtain a quotient by another, we obtain a quotient and a remainder. Thus the and a remainder. Thus the dividend can be written as:dividend can be written as:

(Divisor)(Quotient) + Remainder(Divisor)(Quotient) + Remainder

THEOREM: DIVISION ALGORITHM FOR POLYNOMIALS

THEOREM: DIVISION ALGORITHM FOR POLYNOMIALS

g(x)r(x)

q(x) g(x)f(x)

OROR

f(x) = g(x) q(x) + r(x)f(x) = g(x) q(x) + r(x)

Dividend divisor quotient Dividend divisor quotient remainderremainder

REAL ZEROS OF A POLYNOMIAL FUNCTION

REAL ZEROS OF A POLYNOMIAL FUNCTION

If the divisor is a polynomial of If the divisor is a polynomial of the form x - c where c is a real the form x - c where c is a real number, then the remainder r(x) number, then the remainder r(x) is either the zero polynomial or a is either the zero polynomial or a polynomial of degree 0. Thus, for polynomial of degree 0. Thus, for such divisors, the remainder is such divisors, the remainder is some number R and we may writesome number R and we may write

f(x) = (x - c) q(x) + Rf(x) = (x - c) q(x) + R

REAL ZEROS OF A POLYNOMIAL FUNCTION

REAL ZEROS OF A POLYNOMIAL FUNCTION

If the x variable in the equation of If the x variable in the equation of f(x) gets replaced by the value c, f(x) gets replaced by the value c, thenthen

f(x) = (x - c) q(x) + Rf(x) = (x - c) q(x) + R

f(c) = (c - c) q(x) + Rf(c) = (c - c) q(x) + R

f(c) = Rf(c) = R

REMAINDER THEOREMREMAINDER THEOREM

Let f be a polynomial function. Let f be a polynomial function. If f(x) is divided by x - c, then If f(x) is divided by x - c, then the remainder is f(c).the remainder is f(c).

Ex: Find the remainder if Ex: Find the remainder if

f(x) = xf(x) = x33 - 4x - 4x22 + 2x - 5 is + 2x - 5 is divided divided

by (a) x - 3 and (b) x + 2 by (a) x - 3 and (b) x + 2

FACTOR THEOREMFACTOR THEOREM

Let f be a polynomial function. Let f be a polynomial function. Then x - c is a factor of f(x) if Then x - c is a factor of f(x) if and only if f(c) = 0.and only if f(c) = 0.

Ex: Use the Factor Theorem to Ex: Use the Factor Theorem to determine whether the determine whether the

function function

f(x) = 2xf(x) = 2x33 - x - x22 + 2x - 3 has + 2x - 3 has the the factor (a) x - 1 and factor (a) x - 1 and (b) x + 3 (b) x + 3

THEOREM: NUMBER OF ZEROS

THEOREM: NUMBER OF ZEROS

A polynomial function cannot A polynomial function cannot have more zeros than its have more zeros than its degree.degree.

DESCARTES’ RULE OF SIGNS

DESCARTES’ RULE OF SIGNS

Let f denote a polynomial Let f denote a polynomial function.function.

The number of positive real zeros The number of positive real zeros of f either equals the number of of f either equals the number of variations in sign of the nonzero variations in sign of the nonzero coefficients of f(x) or else equals coefficients of f(x) or else equals that number less an even that number less an even integer.integer.

DESCARTES’ RULE OF SIGNS

DESCARTES’ RULE OF SIGNS

Let f denote a polynomial Let f denote a polynomial function.function.

The number of negative real The number of negative real zeros of f either equals the zeros of f either equals the number of variations in sign of number of variations in sign of the nonzero coefficients of f(- x) the nonzero coefficients of f(- x) or else equals that number less or else equals that number less an even integer.an even integer.

EXAMPLEEXAMPLE

Discuss the real zeros of Discuss the real zeros of

f(x) = 3xf(x) = 3x66 - 4x - 4x44 + 3x + 3x33 + 2x + 2x22 - x - - x - 33

RATIONAL ZEROS THEOREM

RATIONAL ZEROS THEOREM

Let f be a polynomial function of Let f be a polynomial function of degree 1 or higher of the formdegree 1 or higher of the form

f(x) = a f(x) = a n n x x nn + a + a n-1n-1 x x n-1n-1 + . . . + a + . . . + a11x x + a+ a00

(a(an n 0, a 0, a00 0) where each 0) where each coefficient is an integer. If p/q, in coefficient is an integer. If p/q, in lowest terms, is a rational zero of lowest terms, is a rational zero of f, then p must be a factor of af, then p must be a factor of a00 and q must be a factor of aand q must be a factor of ann..

EXAMPLEEXAMPLE

f(x) = 2xf(x) = 2x22 - x - 3 - x - 3

(2x - 3)(x + 1)(2x - 3)(x + 1)

zeros: 3/2, - 1zeros: 3/2, - 1

EXAMPLEEXAMPLE

List the potential rational List the potential rational zeros of f(x) = 2xzeros of f(x) = 2x33 + 11x + 11x22 - 7x - - 7x - 66

p: p: 1, 1, 2, 2, 3, 3, 6 6

q:q: 1, 1, 2 2

FINDING THE REAL ZEROS OF A POLYNOMIAL

FUNCTION

FINDING THE REAL ZEROS OF A POLYNOMIAL

FUNCTION

EXAMPLES 5, 6 & 7EXAMPLES 5, 6 & 7

THEOREMTHEOREM

Every polynomial function (with Every polynomial function (with real coefficients) can be uniquely real coefficients) can be uniquely factored into a product of linear factored into a product of linear factors and/or irreducible factors and/or irreducible quadratic factors.quadratic factors.

COROLLARYCOROLLARY

Every polynomial function (with Every polynomial function (with real coefficients) of odd degree real coefficients) of odd degree has at least one real zero.has at least one real zero.

BOUNDS ON ZEROSBOUNDS ON ZEROSBOUNDS ON ZEROSBOUNDS ON ZEROS

We won’t worry about this We won’t worry about this topic.topic.

Intermediate Value Intermediate Value TheoremTheorem

Intermediate Value Intermediate Value TheoremTheorem

Let Let f f denote a continuous denote a continuous function. If function. If aa < < bb and if and if f f ((aa) ) and and f f ((bb) are of opposite ) are of opposite signs, then the graph of signs, then the graph of ff has has at least one x-intercept at least one x-intercept between between aa and and bb..

EXAMPLEEXAMPLEEXAMPLEEXAMPLE

Use the Intermediate Value Use the Intermediate Value Theorem to show that the Theorem to show that the graph of the function has an graph of the function has an x-intercept in the given x-intercept in the given interval. Approximate the x-interval. Approximate the x-intercept correct to 2 decimal intercept correct to 2 decimal places.places.

f(x) = xf(x) = x44 + 8x + 8x33 - x - x22 + 2; [- 1, + 2; [- 1, 0]0]

f(x) = x f(x) = x 44 + 8x + 8x 33 - x - x 22 + 2; + 2; [- 1, 0][- 1, 0]

f(x) = x f(x) = x 44 + 8x + 8x 33 - x - x 22 + 2; + 2; [- 1, 0][- 1, 0]

f(-1) = (- 1)f(-1) = (- 1)44 + 8(- 1) + 8(- 1)33 - (- 1) - (- 1)22 + 2 + 2

= 1 - 8 - 1 + 2= 1 - 8 - 1 + 2

= - 6= - 6

f(0) = (0)f(0) = (0)44 + 8(0) + 8(0)33 - (0) - (0)22 + 2 + 2

= 0 + 0 - 0 + 2= 0 + 0 - 0 + 2

= 2= 2

EXAMPLEEXAMPLEEXAMPLEEXAMPLE

Use the IVT to show that the Use the IVT to show that the graph of the function has an x-graph of the function has an x-intercept in the given interval. intercept in the given interval. Approximate the x-intercept Approximate the x-intercept correct to 2 decimal places.correct to 2 decimal places.

f(x) = x f(x) = x 55 - 3x - 3x 44 - 2x - 2x 33 +6x +6x 22 + x + 2; + x + 2; [1.7,1.8][1.7,1.8]

Use your calculator for f(1.7) and Use your calculator for f(1.7) and f(1.8).f(1.8).

CONCLUSION OF SECTION 3.5CONCLUSION OF SECTION 3.5