3. Find the zeros of the polynomial x2 – 3x – m(m + 3).
Answers
ANSWER
Given: equation x
2
−3x−m(m+3)=0, where m is a constant
To find the roots of the equation
Sol: the given equation is of form ax
2
+bx+c=0∴a=1,b=−3,c=−m(m+3)
We know the roots of the equation can be find out using the formula,
x=
2a
−b±
b
2
−4ac
Substituting the values of a, b, c, we get
x=
2
−(−3)±
(−3)
2
−4(1)(−m(m+3)
⟹x=
2
3±
9+4m
2
+12m
⟹x=
2
3±(2m+3)
or x=
2
3+(2m+3)
,x=
2
3−(2m+3)
⟹x=m+3,x=−m are the required roots of the equation.
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f(x) = x2 – 3x – m (m + 3)
f(x) = x2 – 3x – m (m + 3) By adding and subtracting mx, we get
f(x) = x2 – 3x – m (m + 3) By adding and subtracting mx, we get f(x) = x2 – mx – 3x + mx – m (m + 3)
f(x) = x2 – 3x – m (m + 3) By adding and subtracting mx, we get f(x) = x2 – mx – 3x + mx – m (m + 3) = x[x – (m + 3)] + m[x – (m + 3)]
f(x) = x2 – 3x – m (m + 3) By adding and subtracting mx, we get f(x) = x2 – mx – 3x + mx – m (m + 3) = x[x – (m + 3)] + m[x – (m + 3)] = [x – (m + 3)] (x + m)
f(x) = x2 – 3x – m (m + 3) By adding and subtracting mx, we get f(x) = x2 – mx – 3x + mx – m (m + 3) = x[x – (m + 3)] + m[x – (m + 3)] = [x – (m + 3)] (x + m) f(x) = 0 ⇒ [x – (m + 3)] (x + m) = 0
f(x) = x2 – 3x – m (m + 3) By adding and subtracting mx, we get f(x) = x2 – mx – 3x + mx – m (m + 3) = x[x – (m + 3)] + m[x – (m + 3)] = [x – (m + 3)] (x + m) f(x) = 0 ⇒ [x – (m + 3)] (x + m) = 0 ⇒ [x – (m + 3)] = 0 or (x + m) = 0
f(x) = x2 – 3x – m (m + 3) By adding and subtracting mx, we get f(x) = x2 – mx – 3x + mx – m (m + 3) = x[x – (m + 3)] + m[x – (m + 3)] = [x – (m + 3)] (x + m) f(x) = 0 ⇒ [x – (m + 3)] (x + m) = 0 ⇒ [x – (m + 3)] = 0 or (x + m) = 0 ⇒ x = m + 3 or x = –m
So, the zeroes of f(x) are –m and +3