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Please log on to your computers. Algebra II Sect 2.7 Please log on to your computers.

Write a formula for the volume of the box below:

Write a formula for the volume of the box below: If the volume is equal to 2730 cubic units, what are the dimensions of the box?

GUIDED PRACTICE How many solutions does the equation x4 + 5x2 – 36 = 0 have? ANSWER 4

Use A Calculator! GUIDED PRACTICE How many solutions does the equation x4 + 5x2 – 36 = 0 have? What are they? ANSWER 4

PRACTICE Find the number of solutions or zeros How many zeros does the function f (x) = x4 – 8x3 + 18x2 – 27 have?

Use A Calculator! PRACTICE How many zeros does the function f (x) = x4 – 8x3 + 18x2 – 27 have? What are they?

PRACTICE Find the number of solutions or zeros How many zeros does the function f (x) = x4 – 8x3 + 18x2 – 27 have? What are the solutions? SOLUTION Because f (x) = x4 – 8x3 + 18x2 – 27 is a polynomial function of degree 4, it has four zeros. (The zeros are – 1, 3, 3, and 3.)

PRACTICE Find the number of solutions or zeros How many solutions does the equation x3 + 5x2 + 4x + 20 = 0 have? SOLUTION Because x3 + 5x2 + 4x + 20 = 0 is a polynomial equation of degree 3,it has three solutions. (The solutions are – 5, – 2i, and 2i.)

EXAMPLE No Calculator! Find all zeros of f (x) = x5 – 4x4 + 4x3 + 10x2 – 13x – 14. SOLUTION STEP 1 Find the rational zeros of f. Because f is a polynomial function of degree 5, it has 5 zeros. The possible rational zeros are + 1, + 2, + 7, and + 14. Using synthetic division, you can determine that – 1 is a zero repeated twice and 2 is also a zero. STEP 2 Write f (x) in factored form. Dividing f (x) by its known factors x + 1, x + 1, and x – 2 gives a quotient of x2 – 4x + 7. Therefore: f (x) = (x + 1)2(x – 2)(x2 – 4x + 7)

EXAMPLE Find the zeros of a polynomial function STEP 3 Find the complex zeros of f . Use the quadratic formula to factor the trinomial into linear factors. f(x) = (x + 1)2(x – 2) x – (2 + i 3 ) x – (2 – i 3 ) The zeros of f are – 1, – 1, 2, 2 + i 3 , and 2 – i 3. ANSWER

GUIDED PRACTICE Find all zeros of the polynomial function. 1. f (x) = x3 + 7x2 + 15x + 9 The zeros of f are – 1, −3, and – 3. ANSWER 2. f (x) = x5 – 2x4 + 8x2 – 13x + 6 ANSWER Zeros of f are 1, 1, – 2, 1 + i 2, and 1 – i 2