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Solving Polynomial Equations

Solving Polynomial Equations. PPT 5.3.2. Factor Polynomial Expressions. In the previous lesson, you factored various polynomial expressions. Such as: x 3 – 2x 2 = x 4 – x 3 – 3x 2 + 3x = = =.

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Solving Polynomial Equations

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  1. Solving Polynomial Equations PPT 5.3.2

  2. Factor Polynomial Expressions In the previous lesson, you factored various polynomial expressions. Such as: x3 – 2x2 = x4 – x3 – 3x2 + 3x = = = Grouping – common factor the first two terms and then the last two terms. Refer to 5.2.2 in Lesson 2 to review which strategy is required for each question. Common Factor x2(x – 2) x(x3 – x2 – 3x + 3) x[x2(x – 1) – 3(x – 1)] Common Factor x(x2 – 3)(x – 1)

  3. Solving Polynomial Equations The expressions on the previous slide are now equations: y = x3 – 2x2 and y = x4 – x3 – 3x2 +3x To solve these equations, we will be solving for x when y = 0.

  4. Solve y = x3 – 2x2 0 = x3 – 2x2 0 = x2(x – 2) x2 = 0 or x – 2 = 0 x = 0 x = 2 Therefore, the roots are 0 and 2. Let y = 0 Common factor Separate the factors and set them equal to zero. Solve for x

  5. Solve Let y = 0 y = x4 – x3 – 3x2 + 3x 0 = x4 – x3 – 3x2 + 3x 0 = x(x3 – x2 – 3x + 3) 0 =x[x2(x – 1) – 3(x – 1)] 0 = x(x – 1)(x2 – 3) x = 0 or x – 1 = 0 or x2 – 3 = 0 x = 0 x = 1 x = Therefore, the roots are 0, 1 and ±1.73 Common factor Group Separate the factors and set them equal to zero. Solve for x

  6. What are you solving for? In the last two slides we solved for x when y = 0, which we call the roots. But what are roots? If you have a graphing calculator follow along with the next few slides to discover what the roots of an equation represent.

  7. What are roots? Press the Y= button on your calculator. Type x3 – 2x2

  8. Press the GRAPH button. Look at where the graph is crossing the x-axis. The x-intercepts are 0 and 2. If you recall, when we solved for the roots of the equation y = x3 – 2x2, we found them to be 0 and 2. Don’t forget, we also put 0 in for y, so it makes sense that the roots would be the x-intercepts.

  9. Use your graphing calculator to graph the other equation we solved, y = x4 – x3 – 3x2 + 3x As you would now expect, the roots that we found earlier, 0, 1 and ±1.73, are in fact the x-intercepts of the graph.

  10. The Quadratic Formula For equations in quadratic form: ax2 + bx + c = 0, we can use the quadratic formula to solve for the roots of the equation. This equation is normally used when factoring is not an option.

  11. Remember, the root 0 came from an earlier step. Using the Quadratic Formula Solve the following cubic equation: y = x3 + 5x2 – 9x 0 = x(x2 + 5x – 9) x = 0 x2 + 5x – 9 = 0 We can, however, use the quadratic formula. Can this equation be factored? We still need to solve for x here. Can this equation be factored? YES it can – common factor. No. There are no two integers that will multiply to -9 and add to 5. a = 1 b = 5 c = -9 Therefore, the roots are 0, 6.41 and -1.41.

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