Question

1. The polynomial (x³ - 2x² + x + 1) is divided by (x - 1). What is the remainder?

a) 1
b) 0
c) 3
d) -1

2. Write the coefficient of (x²) in the following:
a) (2 + x² + x)
b) (2 - 4x² + x)

3. Verify whether the following are the zeroes of the polynomial (P(x) = 3x - 1):
a) (x = 1)
b) (P(x) = 3x - 1, , x = √3, , -√3)


4. Using Factor Theorem, determine whether (g(x)) is a factor of (p(x)):
i) (P(x) = x³ - 4x² + x + 6, , g(x) = x - 3)
ii) (P(x) = 2x³ - 11x² - 4x + 1, , g(x) = 2x + 1)



5. Find the remainder when (x³ - ax² + 6x - a) is divided by (x - a).

a) (2a)
b) (3a)
c) (4a)
d) (5a)

6. Use suitable identities to find the products:
i) ((x - 4)(x + 10))
ii) ((3x + 4)(3x - 5))
iii) ((-3a + 5b + 4c)²)

a) (x² + 6x - 40), (9x² - 1), (9a² - 25b² + 16c² - 30ab + 24bc - 24ac)
b) (x² + 6x + 40), (9x² + 1), (9a² + 25b² + 16c² - 30ab - 24bc + 24ac)
c) (x² - 6x + 40), (9x² + 1), (9a² - 25b² + 16c² + 30ab + 24bc + 24ac)
d) (x² - 6x - 40), (9x² - 1), (9a² + 25b² + 16c² + 30ab - 24bc - 24ac)

Answer 1

Question 1: The remainder is 1 when the polynomial is divided by (x - 1). Question 2: The coefficient of x² is 1 in (2 + x² + x) and -4 in (2 - 4x² + x). Question 3: The equation P(x) = 3x - 1 does not have (x = 1) as a zero and does not have (x = √3) or (x = -√3) as real zeros.

Step-by-step explanation:

Question 1: To find the remainder when the polynomial (x³ - 2x² + x + 1) is divided by (x - 1), we can use synthetic division. Plugging in 1 for x in the polynomial, we get a remainder of 1. Therefore, the correct answer is a) 1.

Question 2: In the expression (2 + x² + x), the coefficient of x² is 1. In the expression (2 - 4x² + x), the coefficient of x² is -4.

Question 3: To verify whether (x = 1) is a zero of the polynomial P(x) = 3x - 1, we can substitute 1 for x and see if the equation holds true. In this case, P(1) = 3(1) - 1 = 2, which is not equal to zero. Therefore, (x = 1) is not a zero of the polynomial. Additionally, the equation P(x) = 3x - 1 does not have any real zeros at √3 or -√3.