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# Po lynomials approx roots

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# Po lynomials approx roots

Polynomials Roots

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### Po lynomials approx roots

1. 1. Block 2 Polynomials (Approximate Roots)
2. 2. What is to be learned? • How to find approximate roots.
3. 3. x2 + 3x + 1 = 0 Will not factorise → Use Big Nasty Formula x3 + 2x2 – 5 = 0 Using Big L? No whole number root
4. 4. Solving Graphically x2 – 5x + 6 = 0 y = x2 – 5x + 6 2 3 solutions x = 2 , x = 3 solutions occur at y = 0
5. 5. Solving Graphically x3 + 2x2 – 5 = 0 y = x3 + 2x2 – 5 1 2 solution between 1 and 2 1< x < 2 y = 13 + 2(1)2 – 5 = -2 y = 23 + 2(2)2 – 5 = 11 x = 1 x = 2 (1 , -2) (2 , 11)
6. 6. For “exact” root, y = 0 For approximate roots, get as close as you can to y = 0 Looking for x value, when y → 0 Root somewhere between positive and negative value. (of y!) 1 2 (1 , -2) (2 , 11)
7. 7. x = 1.5, x = 1.3, x = 1.2, x = 1.25, x = 1 , x3 + 2x2 – 5 = 0 y = x3 + 2x2 – 5 x = 2 , 1< x < 2 better estimate than x = 2 1< x < 1.5 1< x < 1.3 1.2< x < 1.3 1.2< x < 1.25 calculate x (to 1 d.p.) x = 1.2 to 1 d.p. y = 2.875y = 2.875 y = 0.577y = 0.577 y =y = --0.3920.392 y = 0.078y = 0.078 y = -2y = -2 y = 11y = 11
8. 8. Finding Approximate roots of Polynomials For “exact” root, y = 0 For approximate roots, get as close as you can to y = 0 Looking for x value, when y → 0 Root (x =) between positive and negative value of y.
9. 9. x = 2.5, x = 2.3, x = 2.2, x = 2.25, x = 2 , y = -2 x3 – x2 – 6 = 0 y = x3 – x2 – 6 x = 3 , y = 12 2< x < 3 better estimate than x = 3 2< x < 2.5 2< x < 2.3 2.2< x < 2.3 2.2< x < 2.25 calculate x (to 1 d.p.) x = 2.2 to 1 d.p. root between x = 2 and x = 3 y = 3.375y = 3.375 y = 0.877y = 0.877 y =y = --0.3920.392 y = 0.078y = 0.078 (negative) (positive)
10. 10. x = 1.5, x = 1.7, x = 1.6, x = 1.65, x = 1 , x3 – x2 – 2 = 0 y = x3 – x2 – 2 x = 2 , 1< x < 2 1.5 < x < 2 1.5 < x < 1.7 1.6< x < 1.7 1.65< x < 1.7 calculate x (to 1 d.p.) x = 1.7 to 1 d.p. y = -0.875y = -0.875 y = 0.023y = 0.023 y =y = --0.4640.464 y = -0.230y = -0.230 y = -2y = -2 y = 2y = 2 has a root between x = 1 and x = 2 Key Question