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Ex.1 Find the zeros of the quadratic polynomial x^{2} + 7x + 12, and verify the relation between the zeros and its coefficients.
Sol. We have,
f(x) = x^{2} + 7x + 12 = x^{2} + 4x + 3x + 12
f(x) = x(x + 4) + 3 (x + 4)
f(x) = (x + 4) (x + 3)
The zeros of f(x) are given by f(x) = 0
x^{2} + 7x + 12 = 0
(x + 4) (x + 3) = 0
x + 4 = 0 or, x + 3 = 0
x = – 4 or x = – 3
Thus, the zeros of f(x) = x^{2} + 7x + 12 are α = – 4 and β = – 3
Now, sum of the zeros =α+β = (–4) + (–3) = – 7
Ex.2 Find the zeros of the quadratic polynomial f(x) = abx^{2 } + (b^{2} + ac) x + bc and verify the relationship between the zeros and its coefficients.
Sol. f(x) = abx^{2 } + (b^{2} + ac) x + bc = abx^{2 } + b^{2}x + acx + bc
= bx (ax + b) + c (ax + b) = (ax + b) (bx + c)
So, the value of f(x) is zero when ax + b = 0 or bx + c = 0, i.e. x = –b/a or x = –c/b
Therefore, –b/a and –c/b are the zeros (or roots) of f(x).
Ex.3 Find a quadratic polynomial each with the given numbers as the sum and product of its zeros respectively.
(i) 14 , –1 (ii) 2 , 13
Sol. We know that a quadratic polynomial when the sum and product of its zeros are given is given by –
f(x) = k {x^{2 } – (Sum of the zeros) x + Product of the zeros}, where k is a constant.
(i) Required quadratic polynomial f(x) is given by f(x) = k (x^{2 } – 1/4 x – 1 )
(ii) Required quadratic polynomial f(x) is given by f(x) = k (x^{2 } – √ x 1/3 )
Ex.4 Divide the polynomial 2x^{2 } + 3x + 1 by the polynomial x + 2 and verify the division algorithm.
Sol. We have
Clearly, quotient = 2x – 1 and remainder = 3
Also, (x + 2) (2x – 1) + 3 = 2x^{2 } + 4x – x – 2 + 3 = 2x^{2 }+ 3x + 1
i.e., 2x^{2 } + 3x + 1 = (x + 2) (2x–1) + 3. Thus, Dividend = Divisor × Quotient + Remainder.
Ex.5 Check whether the polynomial t^{2} – 3 is a factor of the polynomial 2t^{4} + 3t^{3} – 2t^{2} – 9t – 12, by dividing the second polynomial by the first polynomial.
Sol. We have
Since the remainder is zero, therefore, the polynomial t^{2} – 3 is a factor of the polynomial 2t^{4} + 3t^{3} – 2t^{2} – 9t – 12
Ex.6 Find all the zeros of 2x^{4} – 3x^{3} – 3x^{2} + 6x – 2, if you know that two of its zeros are √2 and – √2.
Sol. Let p(x) = 2x^{4} – 3x^{3} – 3x^{2} + 6x – 2 be the given polynomial. Since two zeros are √2 and – √2 so, (x –√2 ) and (x+√2 ) are both factors of the given polynomial p(x).
Also, (x – √2) (x + √2) = (x^{2} – 2) is a factor of the polynomial. Now, we divide the given polynomial by x^{2} – 2.
Hence, all the zeros of the polynomial 2x^{4} – 3x^{3} – 3x^{2} + 6x – 2 are √2 , – √2 , 1 and 1/2
Ex.7 On dividing f(x) = x^{3} – 3x^{2} + x + 2 by a polynomial g(x), the quotient and remainder were x – 2 and – 2x + 4,
respectively. Find g(x).
Sol. Here, Dividend = x^{3} – 3x^{2} + x + 2
Quotient = x – 2,
Remainder = – 2x + 4 and Divisor = g(x).
Since Dividend = Divisor × Quotient + Remainder
So, x^{3} – 3x^{2} + x + 2 = g(x) × (x –2) + (–2x + 4)
g(x) × (x – 2) = x^{3} – 3x^{2} + x + 2 + 2x – 4
Ex.8. Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following :
(i) p(x) = x^{3} – 3x^{2 } + 5x – 3, g(x) = x2 – 2
(ii) p(x) = x^{4} – 3x^{2 }+ 4x + 5, g(x) = x^{2 }+ 1 – x
(iii) p(x) = x^{4} – 5x + 6, g(x) = 2 – x^{2 }.
Sol.
Hence, Quotient q(x) = x – 3 and Remainder r(x) = 7x – 9
Ex.9. Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the
first polynomial.
(i) t^{2 }– 3, 2t^{4} + 3t^{3 }– 2t^{2 }– 9t – 12 (ii) x^{2 }+ 3x + 1, 3x^{4} + 5x^{3 }– 7x^{2 }+ 2x + 2
(iii) x^{3} – 3x + 1, x5 – 4x^{3 }+ x^{2 }+ 3x + 1
Sol:
Hence, t^{2}– 3 is a factor of 2t^{4}+ 3t^{3} – 2t^{2} – 9t – 12
Ex.10. Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and
(i) deg p(x) = deg q(x)
(ii) deg q(x) = deg r(x)
(iii) deg r(x) = 0.
Sol. (i) p(x) = 2x^{2} + 2x + 8, g(x) = 2x^{0} = 2; q(x) = x^{2} + x + 4 ; r(x) = 0
(ii) p(x) = 2x^{2} + 2x + 8 ; g(x) = x^{2} + x + 9 ; q(x) = 2 ; r(x) = – 10
(iii) p(x) = x^{3} + x + 5 ; g(x) = x^{2} + 1 ; q(x) = x ; r(x) = 5.
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