When roots of polynomials occur in radical form, they occur as two conjugates.
That is,
The conjugate of (a + √b) is (a - √b) and vice versa.
To show that the given conjugates come from a polynomial, we should create the polynomial from the given factors.
The first factor is x - (a + √b).
The second factor is x - (a - √b).
The polynomial is
f(x) = [x - (a + √b)]*[x - (a - √b)]
= x² - x(a - √b) - x(a + √b) + (a + √b)(a - √b)
= x² - 2ax + x√b - x√b + a² - b
= x² - 2ax + a² - b
This is a quadratic polynomial, as expected.
If you solve the quadratic equation x² - 2ax + a² - b = 0 with the quadratic formula, it should yield the pair of conjugate radical roots.
x = (1/2) [ 2a +/- √(4a² - 4(a² - b)]
= a +/- (1/2)*√(4b)
= a +/- √b
x = a + √b, or x = a - √b, as expected.
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4.) $1.60 per notebook
Answer:
<h2>
min: 52</h2><h2>
max: 78</h2><h2>
median: 63</h2><h2>
lower quartile median: 54</h2><h2>
upper quartile median: 73</h2><h2>
interquartile range: 19</h2>
Step-by-step explanation:
the temperatures ordered from least to greatest
52, 53, 54, 54, 57, 62, 64, 69, 73, 73, 77, 78
the min is 52
the max is 78
median:
52, 53, 54, 54, 57, 62, 64, 69, 73, 73, 77, 78
(62+64)/2=126/2=63
thus the median is 63
now we don't include the median in finding the upper and lower quartile medians:
lower quartile median:
52, 53, 54, 54, 57, 62
54 (same #s no need to add up and divide by 2)
upper quartile median:
64, 69, 73, 73, 77, 78
73 (same #s no need to add up and divide by 2)
interquartile range is:
upper quartile median-lower quartile median:
73-54
19
