Essential Question: How can you determine if x-2 is a factor of a polynomial without factoring?

 Standard Form ◦ Largest Exponent comes first. ◦ Combine like terms (if possible) ◦ The constant (number without a variable) comes last  Example ◦ Write the following polynomials in standard form  -7x + 5x 4   x 2 – 4x + 3x 3 + 2x  5x 4 – 7x 3x 3 + x 2 – 2x

 Degree ◦ The degree of a polynomial is the largest exponent  Example ◦ Find the degree of the polynomial  5x 4 - 7x Quartic Cubic DegreeName Using DegreeExample 0Constant6 1Linearx + 3 2Quadratic3x 2 3Cubic2x 3 – 5x 2 – 2x 4Quarticx 4 + 3x 2 5Quintic-2x 5 + 3x 2 – x + 4 3x 3 + x 2 - 2x

 Number of Terms  Example ◦ Classify each polynomial by the number of terms  5x 4 - 7x Binomial Trinomial Number of Terms Name using Number of Terms Example 1Monomial6 2Binomialx + 3 3Trinomial2x 3 – 5x 2 – 2x More than 3Polynomial of x terms-2x 5 + 3x 2 – x + 4 3x 3 + x 2 - 2x

 Write the expression (x + 1)(x + 2)(x + 3) as a polynomial in standard form. ◦ FOIL the last two terms  (x + 1)(x + 2)(x + 3)  (x + 1)(x 2 + 3x + 2x + 6)  (x + 1)(x 2 + 5x + 6) ◦ Distribute the (x + 1) to all terms  (x + 1)(x 2 + 5x + 6)  x 3 + 5x 2 + 6x + x 2 + 5x + 6 ◦ Combine like terms  x 3 + 6x x + 6

 Y OUR TURN Write the expression (x + 1)(x + 1)(x + 2) as a polynomial in standard form.  (x + 1)(x + 1)(x + 2) (x + 1)(x 2 + 2x + 1x + 2) (x + 1)(x 2 + 3x + 2) x 3 + 3x 2 + 2x + x 2 + 3x + 2 x 3 + 4x 2 + 5x + 2

 Writing a Polynomial in Factored Form ◦ Write 2x x x in factored form ◦ Factor out a GCF first  2x x x  2x(x 2 + 5x + 6) ◦ Factor the quadratic in parenthesis  2x(x + 2)(x + 3)

 Y OUR TURN Write 3x 3 – 3x 2 – 36x in factored form  3x(x 2 – x – 12) 3x(x – 4)(x + 3)

 Assignment ◦ Page 309  Problems 1 – 11 (odd problems)  Make sure to include the original problem ◦ Page 317  Problems 1 – 6 (all problems)  Problems 7 – 11 (odd problems)  Show your work  Remember: (x – 3) 2 means (x – 3)(x – 3), not x 2 + 9

Essential Question: How can you determine if x-2 is a factor of a polynomial without factoring?

 If a polynomial is in factored form, you can use the Zero Product Property to find values that will make the polynomial equal zero.  Example ◦ Find the zeros of y = (x – 2)(x + 1)(x + 3). ◦ Just like factoring, if any of the parenthesis come out as zero, then the function is zero.  x – 2 = 0orx + 1 = 0orx + 3 = 0 x + 2 = +2x – 1 = -1x – 3 = -3  -2 x = 2+1 x = -1+3 x = -3  Your Turn ◦ Find the zeros of the function y = (x – 7)(x – 5)(x – 3) ◦ x = 7, 5, or 3

 Writing a Polynomial Function From Its Zeros ◦ Write a polynomial function in standard form with zeros at -2, 3 and 3 ◦ Just the opposite of what we did in the last example, except we also have to multiply the factors together  (x + 2)(x – 3)(x – 3)  FOIL the last two terms  (x + 2)(x 2 – 6x + 9)  Distribute the x + 2 to all terms  x 3 – 6x 2 + 9x + 2x 2 – 12x + 18  Combine like terms  x 3 – 4x 2 – 3x + 18

 Y OUR TURN Write a polynomial function in standard form with zeros at -4, -2 and 1  (x + 4)(x + 2)(x - 1) (x + 4)(x 2 – 1x + 2x - 2) (x + 4)(x 2 + x - 2) x 3 + x 2 – 2x + 4x 2 + 4x - 8 x 3 + 5x 2 + 2x – 8

 Multiplicity ◦ Sometimes, a zero can show up multiple times. Though we generally don’t list multiple zeros as solutions, a multiple zero has MULTIPLICITY equal to the number of times the zero occurs. ◦ Example:  f(x) = x 4 + 6x 3 + 8x 2  f(x) = x 2 (x 2 + 6x + 8)  f(x) = x 2 (x + 4)(x + 2)  Note: you can rewrite x 2 as (x – 0) 2 or (x – 0)(x – 0)  The zeros are x = 0 (multiplicity 2), x = -4, and x = -2

FFind the zeros of the function. State any multiplicity of multiple zeros. ◦f◦f(x) = (x – 2)(x - 1)(x + 1) 2 xx = 2, x = 1, x = -1 (multiplicity 2) ◦y◦y = x 3 – 4x 2 + 4x yy = x(x 2 – 4x + 4) yy = x(x – 2)(x – 2) xx = 0, x = 2 (multiplicity 2)

 Assignment ◦ Page  Problems 16 – 27 (all problems)  Problems 29 – 35, 41 – 45 (odd problems)  Show your work (not relevant in 16-20)  Don’t graph problems