6-2 Polynomials and 6-2 Polynomials and Linear FactorsLinear Factors

Standard and Factored FormStandard and Factored Form

Standard form means to write it as a Standard form means to write it as a simplified (multiplied out) polynomial simplified (multiplied out) polynomial starting with the highest degree term and starting with the highest degree term and working down to the constant term.working down to the constant term.

3x3x22 + 2x – 7 + 2x – 7 Factored form means to write it as the Factored form means to write it as the

product of two or more factors by product of two or more factors by factoring. (remember GCF first!)factoring. (remember GCF first!)

(x – 4)(x + 2)(x + 1)(x – 4)(x + 2)(x + 1)

ZerosZeros

A A zerozero is a (solution or x-intercept) to a is a (solution or x-intercept) to a polynomial function. polynomial function.

If If (x – (x – aa)) is a factor of a polynomial, then is a factor of a polynomial, then aa is a zero (solution) of the function. is a zero (solution) of the function.

If a polynomial has a repeated solution, it If a polynomial has a repeated solution, it has a has a multiple zeromultiple zero..

The number of repeats of a zero is called The number of repeats of a zero is called its its multiplicitymultiplicity..

Finding ZerosFinding Zeros

To find the zeros of a polynomial, we can To find the zeros of a polynomial, we can either graph or we can factor.either graph or we can factor.

(2x – 1)(2x – 1)(x + 5)(x – 5)(2x – 1)(2x – 1)(x + 5)(x – 5)

Finding ZerosFinding Zeros

To find the zeros of a polynomial, we can To find the zeros of a polynomial, we can either graph or we can factor.either graph or we can factor.

(2x – 1)(2x – 1)(2x – 1)(x + 5)(x – 5)(2x – 1)(x + 5)(x – 5)

2x – 1=02x – 1=0

Finding ZerosFinding Zeros

To find the zeros of a polynomial, we can To find the zeros of a polynomial, we can either graph or we can factor.either graph or we can factor.

(2x – 1)(2x – 1)(2x – 1)(x + 5)(x – 5)(2x – 1)(x + 5)(x – 5)

2x =12x =1

Finding ZerosFinding Zeros

To find the zeros of a polynomial, we can To find the zeros of a polynomial, we can either graph or we can factor.either graph or we can factor.

(2x – 1)(2x – 1)(2x – 1)(x + 5)(x – 5)(2x – 1)(x + 5)(x – 5)

x = ½ x = ½

Finding ZerosFinding Zeros

To find the zeros of a polynomial, we can To find the zeros of a polynomial, we can either graph or we can factor.either graph or we can factor.

(2x – 1)(2x – 1)(2x – 1)(2x – 1)(x + 5)(x – 5)(x + 5)(x – 5)

x = ½ (multiplicity 2)x = ½ (multiplicity 2)

Finding ZerosFinding Zeros

To find the zeros of a polynomial, we can To find the zeros of a polynomial, we can either graph or we can factor.either graph or we can factor.

(2x – 1)(2x – 1)(2x – 1)(2x – 1)(x + 5)(x + 5)(x – 5)(x – 5)

x = ½x = ½

Finding ZerosFinding Zeros

To find the zeros of a polynomial, we can To find the zeros of a polynomial, we can either graph or we can factor.either graph or we can factor.

(2x – 1)(2x – 1)(2x – 1)(2x – 1)(x + 5)(x + 5)(x – 5)(x – 5)

x = ½x = ½, -5, -5

Finding ZerosFinding Zeros

To find the zeros of a polynomial, we can To find the zeros of a polynomial, we can either graph or we can factor.either graph or we can factor.

(2x – 1)(2x – 1)(x + 5)(2x – 1)(2x – 1)(x + 5)(x – 5)(x – 5)

x = ½, -5x = ½, -5

Finding ZerosFinding Zeros

To find the zeros of a polynomial, we can To find the zeros of a polynomial, we can either graph or we can factor.either graph or we can factor.

(2x – 1)(2x – 1)(x + 5)(2x – 1)(2x – 1)(x + 5)(x – 5)(x – 5)

x = ½, -5x = ½, -5, 5, 5

RememberRemember

The following are equivalent statements:The following are equivalent statements: -4-4 is a is a solutionsolution of x of x22 + 3x – 4 + 3x – 4

-4-4 is an is an x-interceptx-intercept of x of x22 + 3x – 4 + 3x – 4 -4-4 is a is a zerozero of y = x of y = x22 + 3x – 4 + 3x – 4 x + 4x + 4 is a is a factorfactor of x of x22 + 3x – 4 + 3x – 4