write the coefficient of x^2 of the following 8x^3-20x^3+3x+5
Answers
(i) f(x) = x2 − 2x − 8
(ii) g(s) = 4s2 − 4s + 1
(iii) h(t) = t2 − 15
(iv) 6x2 − 3 − 7x
(v) p(x)=x2+22–√x−6
(vi) q(x)=3–√x2+10x+73–√
(vii)f(x)=x2−(3–√+1) x+3–√
(viii) g(x) = a(x2 + 1) − x(a2 + 1)
(ix) h(s)=2s2−(1+22–√)s+2–√
(x) f(v)=v2+43–√v−15
(xi) p(y)=y2+35√2y−5
(xii) q(y)=7y2−113y−23
ANSWER:
(i) We have,
f(x) = x2 − 2x − 8
f(x) = x2 + 2x − 4x − 8
f(x) = x (x + 2) − 4(x + 2)
f(x) = (x + 2) (x − 4)
The zeros of f(x) are given by
f(x) = 0
x2 − 2x − 8 = 0
(x + 2) (x − 4) = 0
x + 2 = 0
x = −2
Or
x − 4 = 0
x = 4
Thus, the zeros of f(x) = x2 − 2x − 8 are α = −2 and β = 4
Now,
and
Therefore, sum of the zeros =
Product of the zeros
= − 2 × 4
= −8
and
Therefore,
Product of the zeros =
Hence, the relation-ship between the zeros and coefficient are verified.
(ii) Given
When have,
g(s) = 4s2 − 4s + 1
g(s) = 4s2 − 2s − 2s + 1
g(s) = 2s (2s − 1) − 1(2s − 1)
g(s) = (2s − 1) (2s − 1)
The zeros of g(s) are given by
Or
Thus, the zeros ofare
and
Now, sum of the zeros
and
Therefore, sum of the zeros =
Product of the zeros
and =
Therefore, the product of the zeros =
Hence, the relation-ship between the zeros and coefficient are verified.
(iii) Given
We have,
h(t) = t2 − 15h(t) = (t)2 − (15−−√)2h(t) = (t + 15−−√) (t − 15−−√)
The zeros of are given by
h(t) = 0(t − 15−−√) (t + 15−−√) = 0(t − 15−−√) = 0t = 15−−√or (t + 15−−√) =