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Algebra II Section 2.7: Polynomial Zeros and Solutions

Learn how to find the number of solutions and zeros of polynomial equations and functions. Use a calculator or synthetic division to find rational or complex zeros, and factor the polynomial into linear factors.

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Algebra II Section 2.7: Polynomial Zeros and Solutions

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  1. Algebra IISect 2.7 Please log on to your computers.

  2. Write a formula for the volume of the box below:

  3. 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?

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

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

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

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

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

  9. PRACTICE Find the number of solutions or zeros How many solutions does the equation x3 + 5x2 + 4x + 20 = 0have? 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.)

  10. STEP 1 Find the rational zeros of f. Because fis 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: EXAMPLE No Calculator! Find all zeros of f (x) = x5 – 4x4 + 4x3 + 10x2 – 13x – 14. SOLUTION f (x) = (x + 1)2(x – 2)(x2 – 4x + 7)

  11. 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 ) ANSWER The zeros of f are – 1, – 1, 2, 2 + i 3 , and2 – i 3. EXAMPLE Find the zeros of a polynomial function

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

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