Zeroes of a polynomial | Polynomials | Assessment | English | Grade 9

Zero of the zero polynomial is ______

• 0

• 1

• Any real number

• Not defined

Answer explanation

By convention, every real number is a zero of the zero polynomial.

The value of polynomial x² - 5x + 10 at x = 2 is ____

• 4

• 0

• -4

• 2

Answer explanation

p(x)= x² - 5x + 10 p(2)= (2)² - 5(2) + 10 = 4 - 10 + 10 = 4 Hence value of polynomial x² - 5x + 10 at x = 2 is 4.

Find the zeroes of polynomial of x² - 4x

• 0

• 4

• 0, 4

• 0, -4

Answer explanation

To find the zeroes of the polynomial x² - 4x we will equate it to zero and then solve x² - 4x = 0 x(x - 4) = 0 x = 0 or x - 4 = 0 x = 4 Hence, 0 and 4 are zeroes of polynomial x² - 4x

One of the zeroes of the polynomial 12x³ - 24x² - 4x + 8 is__________________

• -2/√12

• -2

• 1

• 12

Answer explanation

Let p(x)= 12x³ - 24x² - 4x + 8 Evaluate p(x) for the values given in the options p(-2/√12) = (-2/√12)³ - 24 (-2/√12)² - 4(-2/√12) +8 = 0 p(-2) = 12(-2)³ -24(-2)² - 4(-2) + 8 = -176 p(1) =12(1)³ - 24(1)² - 4(1) + 8 = -8 p(12) = 12(12)³ - 24(12)² - 4(12) + 8 = 17,240 Only in Option 1 i.e. p(-2/√12), we are getting the value as zero. Hence -2/√12 is one of the zeroes of 12x³ - 24x² - 4x + 8 Therefore, option 1 is correct Alternatively, find the roots of the equation p(x)=0 12x³ - 24x² - 4x + 8 = 0 12x² (x -2) - 4 (x -2) = 0 (12x² - 4) (x - 2) = 0 (√12x + 2) (√12x - 2) (x - 2) = 0 √12x + 2 = 0 √12x - 2 = 0 x -2 = 0 x = -2/√12 x = 2/√12 x = 2 We can see that option 1 is -2/√12 which is one of the zeroes for the polynomial. Hence, option 1 is correct.

Which of the following statements is false?

• –8 is a zero of 2y + 16

• –3 is a zero of x² + x – 6

• One the zeroes of polynomial x² + 7x + 12 is 4

• The value of polynomial 5x² + 7x + 2 when x = -2/5 is 0

Answer explanation

Option 3 is the correct answer as it is the only false statement in the given options. Let p(x)= x² + 7x + 12 Evaluate p(x) at x = 4 p(4)= (4)² + 7(4) +12 = 16 + 28 +12 = 56 Since p(4)≠ 0, 4 is not a zero of the given polynomial. The given statement is false and is the correct answer for this question. For option 1: –8 is a zero of 2y + 16 Let p(y)= 2y +16 Solve p(y)=0 2y + 16 = 0 2y = -16 y = -16/2 y = -8 -8 is a zero of polynomial 2y +16. hence Option 1 is true and is not the correct answer of this question. For option 2: –3 is a zero of x² + x – 6 Let p(x) = x² + x – 6 Evaluate p(x) at x = -3 p(-3)= (-3)(-3) + (-3) -6 p(-3)= 9 -3 -6 p(-3)= 0 Hence, –3 is a zero of x² + x – 6. Therefore, option 2 is true and is not the correct answer of this question. For option 4: The value of polynomial 5x² + 7x + 2 when x = -2/5 is 0 p(x)= 5x² + 7x + 2 p(-2/5)= 5(-2/5)² + 7(-2/5) + 2 = 5 (4/25) - 14/5 + 2 = 4/5 - 14/5 + 2 = -10/5 + 2 = -10/5 + 10/5 = 0 Therefore, option 4 is true and is not the correct answer of this question.

If p(x) = √4x² + 5x - 7, then evaluate 2p(3) + 9p(⅓) - p(-2)

• 15

• -2/5

• 9

• 0

Answer explanation

Let p(x) = √4x² + 5x - 7 Evaluate p(3), p(⅓) and p(-2) p(3) = √4(3)² + 5(3) - 7 p(3) = 2 x 9 + 15 - 7 p(3) = 26 p(⅓) = √4(⅓)² + 5 (⅓) - 7 p(⅓) = 2 x (⅑) + 5/3 - 7 p(⅓) = 2/9 + 5/3 - 7 p(⅓) = 2/9 + 15/9 - 7 p(⅓) = 17/9 - 7 p(⅓) = -46/9 p(-2) = √4(-2)² + 5(-2) - 7 p(-2) = 2 x 4 - 10 - 7 p(-2) = 8 -17 p(-2) = -9 ∴2p(3) + 9p(⅓) - p(-2) = 2 x 26 + 9 x -46/9 - (-9) ∴2p(3) + 9p(⅓) - p(-2) = 52 - 46 + 9 ∴2p(3) + 9p(⅓) - p(-2) = 15 Therefore, option 1 is the correct answer.