Question

Verify whether the indicated numbers are zeros of the polynomials corresponding to them in the following cases : (i) f(x) = 3x + 1, x = - 13 (ii) f(x) = x2 - 1, x = 1, -1 (iii) g(x) = 3x2 - 2, x = 2√3,−2√3 (vi) p(x) = x3 − 6x2 + 11x - 6, x = 1,2,3 (v) f(x) = 5x -π,x=45 (vi) f(x) = x2,x = 0 (vii) f(x) = lx + m, = - ml (viii) f(x) = 2x + 1, x = 12

Solution

(i) f(x) = 3x + 1, x = - 13 f(−13)=3(−13)+4 = - 1 + 1 = 0 ∴x=−13 is the zero of f(x)  (ii) f(x) = x2 - 1, x=1, -1 f(1) =(1)2 - 1 = 1 -1 =0 x=1 is zero of f(x) f(-1)=(−1)2 - 1 = 1 = 0 x = - 1 is zero of f(x) (iii) g(x) = 3x2 - 2, x = 2√3,−2√3 g(2√3)=3(2√3)2−2=3×43−2 =4-2=2 ∴x=2√3 is not its zero g(2√3)=3(2√3)2−2 =3×43−2=4−2=2 ∴x=−23 is not its zero  (iv) p(x) = x3−6x3 + 11x - 6, x = 1, 2,3 P(1) =(1)3−6(1)2 + 11(1) - 6 = 1 - 6 ×1+11× 1 - 6 = 1 - 6 + 11 - 6 =12 - 12 = 0 ∴ x = 1 is its zero p(2) =(2)2−6(2)2+11× 2 - 6 = 8 - 6× 4 + 22 - 6 = 8 - 24 + 22 - 6 =30 - 30 = 0 ∴ x = 2 is its zero P(3) =(3)3−6(3)2+11× 3 - 6 = 27 - 6× 9 + 33 - 6 = 27 - 54 + 33 - 6 = 60 - 60 = 0 ∴ x = 3 is its zero Hence x = 1,2,3 are its zeros (v) f(x) = 5x - π,x=45 =f(45)5×45−π=4−π ∴x=45 is not its zero (vi) f(x) = x2,x = 0 ∴ f(0)=(0)2 = 0 ∴ x = 0 is its zero (vii) f(x) = lx + m, x = - ml ∴f(−ml)=1×(−ml)+m = - m + m = 0 ∴=−ml is its zero (viii) f(x) = 2x + 1, x =12 ∴f(12)=2×(12)2+1=2×14+1 =12+1=32 ∴x=12 is not its zero MathematicsRD SharmaStandard IX

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