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Block 2
Polynomials(Approximate Roots)
What is to be learned?
• How to find approximate roots.
x2 + 3x + 1 = 0Will not factorise→ Use Big Nasty Formula
x3 + 2x2 – 5 = 0Using Big L?No whole number root
Solving Graphically
x2 – 5x + 6 = 0
y = x2 – 5x + 6
2 3
solutions
x = 2 , x = 3
solutions occur at y = 0
Solving Graphically
x3 + 2x2 – 5 = 0
y = x3 + 2x2 – 5
1 2
solution between 1 and 21< x < 2
y = 13 + 2(1)2 – 5 = -2
y = 23 + 2(2)2 – 5 = 11
x = 1
x = 2(1 , -2)
(2 , 11)
For “exact” root, y = 0For approximate roots, get as close as you can to y = 0Looking for x value, when y → 0Root somewhere between positive and negative value. (of y!)
1 2
(1 , -2)
(2 , 11)
x = 1.5,
x = 1.3, x = 1.2, x = 1.25,
x = 1 ,
x3 + 2x2 – 5 = 0y = x3 + 2x2 – 5
x = 2 , 1< x < 2
better estimate than x = 21< 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 = --0.3920.392y = 0.078y = 0.078
y = -2y = -2y = 11y = 11
Finding Approximate roots of Polynomials
For “exact” root, y = 0For approximate roots, get as close as you can to y = 0Looking for x value, when y → 0Root (x =) between positive and
negative value of y.
x = 2.5,
x = 2.3, x = 2.2, x = 2.25,
x = 2 , y = -2
x3 – x2 – 6 = 0y = x3 – x2 – 6
x = 3 , y = 12 2< x < 3
better estimate than x = 32< 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 = 0.877y = y = --0.3920.392y = 0.078y = 0.078
(negative)
(positive)
x = 1.5, x = 1.7, x = 1.6, x = 1.65,
x = 1 ,
x3 – x2 – 2 = 0y = 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 = -0.875y = 0.023y = 0.023y = y = --0.4640.464y = -0.230y = -0.230
y = -2y = -2y = 2y = 2
has a root between x = 1 and x = 2
Key Question
