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5.2 Polynomials and Linear Factors

5.2 Polynomials and Linear Factors. Linear Factors. Just as you can write a number into its prime factors you can write a poly into its linear factors. Ex. 6 into 2 & 3 x 2 + 4x – 12 into (x+6)(x-2). We can take a poly in factored form and rewrite it into standard form.

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5.2 Polynomials and Linear Factors

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  1. 5.2 Polynomials and Linear Factors

  2. Linear Factors Just as you can write a number into its prime factors you can write a poly into its linear factors. Ex. 6 into 2 & 3 x2 + 4x – 12 into (x+6)(x-2)

  3. We can take a poly in factored form and rewrite it into standard form. Ex. (x+1)(x+2)(x+3) = foil distribute (x2+5x+6)(x+1)=x (x2+5x+6)+1 (x2+5x+6) = x3+6x2+11x+6 Standard form

  4. We can also use the GCF (greatest common factor) to factor a poly in standard form into its linear factors. Ex. 2x3+10x2+12x GCF is 2x so factor it out. We get 2x(x2+5x+6) now factor once more to get 2x(x+2)(x+3) Linear Factors

  5. The greatest y value of the points in a region is called the local maximum. The least y value among nearby points is called the local minimum.

  6. Theorem The expression (x-a) is a linear factor of a polynomial if and only if the value a is a zero (root) of the related polynomial function. If and only if = the theorem goes both ways

  7. Example 1 We can rewrite a polynomial from its zeros. Write a poly with zeros -2,3, and 3 f(x)= (x+2)(x-3)(x-3) foil = (x+2)(x2-6x+9) now distribute to get = x3-4x2-3x+18 this function has zeros at -2,3 and 3

  8. A repeated zero is called a multiple zero. A multiple zero has a multiplicity equal to the number of times the zero occurs. In Example 1, the zero 3 has a multiplicity of 2.

  9. Equivalent Statements about Polys • -4 is a solution of x2+3x-4=0 • -4 is an x-intercept of the graph of y=x2+3x-4 • -4 is a zero of y=x2+3x-4 • (x+4) is a factor of x2+3x-4 These all say the same thing HW pg 293 7-33 odds

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