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Page 1 Exercise 2A 1. Find the zeros of the polynomial f(x) = x 2 + 7x + 12 and verify the relation between its zeroes and coefficients. Sol: x 2 + 7x + 12 = 0 x 2 + 4x + 3x + 12 = 0 x(x+4) + 3(x+4) = 0 (x+4) (x+3) = 0 (x + 4) = 0 or (x + 3) = 0 x = 4 or x = 3 Sum of zeroes = 4 + ( 3) = = Product of zeroes = ( 4) ( 3) = = 2. Find the zeroes of the polynomial f(x) = x 2 and coefficients. Sol: x 2 x 2 x = 4 or x = 2 Sum of zeroes = 4 + ( 2) = 2 = = Product of zeroes = (4) ( 2) = = 3. Find the zeroes of the quadratic polynomial f(x) = x 2 rify the relation between its zeroes and coefficients. Sol: We have: f(x) = x 2 = x 2 = = (x 2) (x + 5) f(x) = 0 (x 2) (x + 5) = 0 x 2 = 0 or x + 5 = 0 x = 2 or x = 5. So, the zeroes of f(x) are 2 and 5. Page 2 Exercise 2A 1. Find the zeros of the polynomial f(x) = x 2 + 7x + 12 and verify the relation between its zeroes and coefficients. Sol: x 2 + 7x + 12 = 0 x 2 + 4x + 3x + 12 = 0 x(x+4) + 3(x+4) = 0 (x+4) (x+3) = 0 (x + 4) = 0 or (x + 3) = 0 x = 4 or x = 3 Sum of zeroes = 4 + ( 3) = = Product of zeroes = ( 4) ( 3) = = 2. Find the zeroes of the polynomial f(x) = x 2 and coefficients. Sol: x 2 x 2 x = 4 or x = 2 Sum of zeroes = 4 + ( 2) = 2 = = Product of zeroes = (4) ( 2) = = 3. Find the zeroes of the quadratic polynomial f(x) = x 2 rify the relation between its zeroes and coefficients. Sol: We have: f(x) = x 2 = x 2 = = (x 2) (x + 5) f(x) = 0 (x 2) (x + 5) = 0 x 2 = 0 or x + 5 = 0 x = 2 or x = 5. So, the zeroes of f(x) are 2 and 5. Sum of zeroes = 2 + ( 5) = 3 = = Product of zeroes = 2 × ( 5) = 10 = = 4. Find the zeroes of the quadratic polynomial f(x) = 4x 2 its zeroes and coefficients. Sol: We have: f(x) = 4x 2 = 4x 2 = 4x 2 3 = 2x (2x 3) + 1(2x 3) = (2x + 1) (2x 3) f(x) = 0 (2x + 1) (2x 3)= 0 2x + 1= 0 or 2x 3 = 0 x = or x = So, the zeroes of f(x) are and . Sum of zeroes = + = = = 1 = Product of zeroes = × = = 5. Find the zeroes of the quadratic polynomial f(x) = 5x 2 between the zeroes and coefficients of the given polynomial. Sol: We have: f(x) = 5x 2 = 5x 2 = 5x 2 = 5x 2 = 5x (x 2) + 2(x 2) = (5x + 2) (x 2) f(x) = 0 (5x + 2) (x 2) = 0 5x + 2= 0 or x 2 = 0 x = or x = 2 So, the zeroes of f(x) are and 2. Sum of zeroes = + 2 = = = Product of zeroes = × 2 = = Page 3 Exercise 2A 1. Find the zeros of the polynomial f(x) = x 2 + 7x + 12 and verify the relation between its zeroes and coefficients. Sol: x 2 + 7x + 12 = 0 x 2 + 4x + 3x + 12 = 0 x(x+4) + 3(x+4) = 0 (x+4) (x+3) = 0 (x + 4) = 0 or (x + 3) = 0 x = 4 or x = 3 Sum of zeroes = 4 + ( 3) = = Product of zeroes = ( 4) ( 3) = = 2. Find the zeroes of the polynomial f(x) = x 2 and coefficients. Sol: x 2 x 2 x = 4 or x = 2 Sum of zeroes = 4 + ( 2) = 2 = = Product of zeroes = (4) ( 2) = = 3. Find the zeroes of the quadratic polynomial f(x) = x 2 rify the relation between its zeroes and coefficients. Sol: We have: f(x) = x 2 = x 2 = = (x 2) (x + 5) f(x) = 0 (x 2) (x + 5) = 0 x 2 = 0 or x + 5 = 0 x = 2 or x = 5. So, the zeroes of f(x) are 2 and 5. Sum of zeroes = 2 + ( 5) = 3 = = Product of zeroes = 2 × ( 5) = 10 = = 4. Find the zeroes of the quadratic polynomial f(x) = 4x 2 its zeroes and coefficients. Sol: We have: f(x) = 4x 2 = 4x 2 = 4x 2 3 = 2x (2x 3) + 1(2x 3) = (2x + 1) (2x 3) f(x) = 0 (2x + 1) (2x 3)= 0 2x + 1= 0 or 2x 3 = 0 x = or x = So, the zeroes of f(x) are and . Sum of zeroes = + = = = 1 = Product of zeroes = × = = 5. Find the zeroes of the quadratic polynomial f(x) = 5x 2 between the zeroes and coefficients of the given polynomial. Sol: We have: f(x) = 5x 2 = 5x 2 = 5x 2 = 5x 2 = 5x (x 2) + 2(x 2) = (5x + 2) (x 2) f(x) = 0 (5x + 2) (x 2) = 0 5x + 2= 0 or x 2 = 0 x = or x = 2 So, the zeroes of f(x) are and 2. Sum of zeroes = + 2 = = = Product of zeroes = × 2 = = 6. Find the zeroes of the polynomial f(x) = 2 + and verify the relation between its zeroes and coefficients. Sol: 2 + 2 + 2 ( ( ( ) = 0 ( ) = 0 x = or x = x = × = or x = Sum of zeroes = + = = Product of zeroes = × = = 7. Find the zeroes of the quadratic polynomial 2x 2 the zeroes and the coefficients. Sol: f(x) = 2x 2 = 2x 2 = 2x 2 = 2x (x x 3) = (2x 5) (x 3) f(x) = 0 (2x 5) (x 3) = 0 2x 5= 0 or x 3 = 0 x = or x = 3 So, the zeroes of f(x) are and 3. Sum of zeroes = + 3 = = = Product of zeroes = × 3 = = 8. Find the zeroes of the quadratic polynomial 4x 2 between the zeroes and the coefficients. Sol: 4x 2 (2x) 2 2(2x)(1) + (1) 2 = 0 Page 4 Exercise 2A 1. Find the zeros of the polynomial f(x) = x 2 + 7x + 12 and verify the relation between its zeroes and coefficients. Sol: x 2 + 7x + 12 = 0 x 2 + 4x + 3x + 12 = 0 x(x+4) + 3(x+4) = 0 (x+4) (x+3) = 0 (x + 4) = 0 or (x + 3) = 0 x = 4 or x = 3 Sum of zeroes = 4 + ( 3) = = Product of zeroes = ( 4) ( 3) = = 2. Find the zeroes of the polynomial f(x) = x 2 and coefficients. Sol: x 2 x 2 x = 4 or x = 2 Sum of zeroes = 4 + ( 2) = 2 = = Product of zeroes = (4) ( 2) = = 3. Find the zeroes of the quadratic polynomial f(x) = x 2 rify the relation between its zeroes and coefficients. Sol: We have: f(x) = x 2 = x 2 = = (x 2) (x + 5) f(x) = 0 (x 2) (x + 5) = 0 x 2 = 0 or x + 5 = 0 x = 2 or x = 5. So, the zeroes of f(x) are 2 and 5. Sum of zeroes = 2 + ( 5) = 3 = = Product of zeroes = 2 × ( 5) = 10 = = 4. Find the zeroes of the quadratic polynomial f(x) = 4x 2 its zeroes and coefficients. Sol: We have: f(x) = 4x 2 = 4x 2 = 4x 2 3 = 2x (2x 3) + 1(2x 3) = (2x + 1) (2x 3) f(x) = 0 (2x + 1) (2x 3)= 0 2x + 1= 0 or 2x 3 = 0 x = or x = So, the zeroes of f(x) are and . Sum of zeroes = + = = = 1 = Product of zeroes = × = = 5. Find the zeroes of the quadratic polynomial f(x) = 5x 2 between the zeroes and coefficients of the given polynomial. Sol: We have: f(x) = 5x 2 = 5x 2 = 5x 2 = 5x 2 = 5x (x 2) + 2(x 2) = (5x + 2) (x 2) f(x) = 0 (5x + 2) (x 2) = 0 5x + 2= 0 or x 2 = 0 x = or x = 2 So, the zeroes of f(x) are and 2. Sum of zeroes = + 2 = = = Product of zeroes = × 2 = = 6. Find the zeroes of the polynomial f(x) = 2 + and verify the relation between its zeroes and coefficients. Sol: 2 + 2 + 2 ( ( ( ) = 0 ( ) = 0 x = or x = x = × = or x = Sum of zeroes = + = = Product of zeroes = × = = 7. Find the zeroes of the quadratic polynomial 2x 2 the zeroes and the coefficients. Sol: f(x) = 2x 2 = 2x 2 = 2x 2 = 2x (x x 3) = (2x 5) (x 3) f(x) = 0 (2x 5) (x 3) = 0 2x 5= 0 or x 3 = 0 x = or x = 3 So, the zeroes of f(x) are and 3. Sum of zeroes = + 3 = = = Product of zeroes = × 3 = = 8. Find the zeroes of the quadratic polynomial 4x 2 between the zeroes and the coefficients. Sol: 4x 2 (2x) 2 2(2x)(1) + (1) 2 = 0 (2x 1) 2 = 0 [ a 2 2ab + b 2 = (a b) 2 ] (2x 1) 2 = 0 x = or x = Sum of zeroes = + = 1 = = Product of zeroes = × = = 9. Find the zeroes of the quadratic polynomial (x 2 ) and verify the relation between the zeroes and the coefficients. Sol: We have: f(x) = x 2 It can be written as x 2 = = (x + ) (x ) f(x) = 0 (x + ) (x ) = 0 x + = 0 or x = 0 x = or x = So, the zeroes of f(x) are and . Here, the coefficient of x is 0 and the coefficient of is 1. Sum of zeroes = + = = Product of zeroes = × = = 10. Find the zeroes of the quadratic polynomial (8x 2 ) and verify the relation between the zeroes and the coefficients. Sol: We have: f(x) = 8x 2 It can be written as 8x 2 = 4 { (1) 2 } = 4 ( + 1) ( 1) f(x) = 0 ( + 1) ( 1) = 0 ( + 1) = 0 or 1 = 0 x = or x = Page 5 Exercise 2A 1. Find the zeros of the polynomial f(x) = x 2 + 7x + 12 and verify the relation between its zeroes and coefficients. Sol: x 2 + 7x + 12 = 0 x 2 + 4x + 3x + 12 = 0 x(x+4) + 3(x+4) = 0 (x+4) (x+3) = 0 (x + 4) = 0 or (x + 3) = 0 x = 4 or x = 3 Sum of zeroes = 4 + ( 3) = = Product of zeroes = ( 4) ( 3) = = 2. Find the zeroes of the polynomial f(x) = x 2 and coefficients. Sol: x 2 x 2 x = 4 or x = 2 Sum of zeroes = 4 + ( 2) = 2 = = Product of zeroes = (4) ( 2) = = 3. Find the zeroes of the quadratic polynomial f(x) = x 2 rify the relation between its zeroes and coefficients. Sol: We have: f(x) = x 2 = x 2 = = (x 2) (x + 5) f(x) = 0 (x 2) (x + 5) = 0 x 2 = 0 or x + 5 = 0 x = 2 or x = 5. So, the zeroes of f(x) are 2 and 5. Sum of zeroes = 2 + ( 5) = 3 = = Product of zeroes = 2 × ( 5) = 10 = = 4. Find the zeroes of the quadratic polynomial f(x) = 4x 2 its zeroes and coefficients. Sol: We have: f(x) = 4x 2 = 4x 2 = 4x 2 3 = 2x (2x 3) + 1(2x 3) = (2x + 1) (2x 3) f(x) = 0 (2x + 1) (2x 3)= 0 2x + 1= 0 or 2x 3 = 0 x = or x = So, the zeroes of f(x) are and . Sum of zeroes = + = = = 1 = Product of zeroes = × = = 5. Find the zeroes of the quadratic polynomial f(x) = 5x 2 between the zeroes and coefficients of the given polynomial. Sol: We have: f(x) = 5x 2 = 5x 2 = 5x 2 = 5x 2 = 5x (x 2) + 2(x 2) = (5x + 2) (x 2) f(x) = 0 (5x + 2) (x 2) = 0 5x + 2= 0 or x 2 = 0 x = or x = 2 So, the zeroes of f(x) are and 2. Sum of zeroes = + 2 = = = Product of zeroes = × 2 = = 6. Find the zeroes of the polynomial f(x) = 2 + and verify the relation between its zeroes and coefficients. Sol: 2 + 2 + 2 ( ( ( ) = 0 ( ) = 0 x = or x = x = × = or x = Sum of zeroes = + = = Product of zeroes = × = = 7. Find the zeroes of the quadratic polynomial 2x 2 the zeroes and the coefficients. Sol: f(x) = 2x 2 = 2x 2 = 2x 2 = 2x (x x 3) = (2x 5) (x 3) f(x) = 0 (2x 5) (x 3) = 0 2x 5= 0 or x 3 = 0 x = or x = 3 So, the zeroes of f(x) are and 3. Sum of zeroes = + 3 = = = Product of zeroes = × 3 = = 8. Find the zeroes of the quadratic polynomial 4x 2 between the zeroes and the coefficients. Sol: 4x 2 (2x) 2 2(2x)(1) + (1) 2 = 0 (2x 1) 2 = 0 [ a 2 2ab + b 2 = (a b) 2 ] (2x 1) 2 = 0 x = or x = Sum of zeroes = + = 1 = = Product of zeroes = × = = 9. Find the zeroes of the quadratic polynomial (x 2 ) and verify the relation between the zeroes and the coefficients. Sol: We have: f(x) = x 2 It can be written as x 2 = = (x + ) (x ) f(x) = 0 (x + ) (x ) = 0 x + = 0 or x = 0 x = or x = So, the zeroes of f(x) are and . Here, the coefficient of x is 0 and the coefficient of is 1. Sum of zeroes = + = = Product of zeroes = × = = 10. Find the zeroes of the quadratic polynomial (8x 2 ) and verify the relation between the zeroes and the coefficients. Sol: We have: f(x) = 8x 2 It can be written as 8x 2 = 4 { (1) 2 } = 4 ( + 1) ( 1) f(x) = 0 ( + 1) ( 1) = 0 ( + 1) = 0 or 1 = 0 x = or x = So, the zeroes of f(x) are and Here the coefficient of x is 0 and the coefficient of x 2 is Sum of zeroes = + = = = Product of zeroes = × = = = 11. Find the zeroes of the quadratic polynomial (5y 2 + 10y) and verify the relation between the zeroes and the coefficients. Sol: We have, f (u) = 5u 2 + 10u It can be written as 5u (u+2) f (u) = 0 5u = 0 or u + 2 = 0 u = 0 or u = 2 So, the zeroes of f (u) are 2 and 0. Sum of the zeroes = 2 + 0 = 2 = = = Product of zeroes = 2 × 0 = 0 = = = 12. Find the zeroes of the quadratic polynomial (3x 2 ) and verify the relation between the zeroes and the coefficients. Sol: 3x 2 3x 2 x (3x 4) + 1 (3x 4) = 0 (3x 4) (x + 1) = 0 (3x 4) or (x + 1) = 0 x = or x = 1 Sum of zeroes = + (1) = = Product of zeroes = × (1) = = 13. Find the quadratic polynomial whose zeroes are 2 and 6. Verify the relation between the coefficients and the zeroes of the polynomial. Sol: Let = 2 and = 6 Sum of the zeroes, ( + ) = 2 + (6) = 4Read More
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1. What are polynomials and how are they used in mathematics? 
2. How do you determine the degree of a polynomial? 
3. What is the factor theorem and how is it used to find the factors of a polynomial? 
4. Can a polynomial have more than one factor? 
5. How do you add and subtract polynomials? 
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