Practice Test: Polynomials

The correct answer is b

Given quadratic polynomial is,

p(x)=x2+ax+b

Let one of the zero of the polynomial is a then the other zero will be −a.

Sum of zeros=a+(−a)= −a / 1

=>0= −a/1

=>a=0

and product of the zeros =a(−a)=b/1

=>−a2=b

=>b=−a2

=>b has to be negative.

Let p(x) =ax3 + bx2 + cx + d

Given that, one of the zeroes of the cubic polynomial p(x) is zero.

Let α, β and γ are the zeroes of cubic polynomial p(x), where α = 0.

We know that,

Step-by-step explanation:

sum of two zeros at a time = c/a

                      ∴αβ + βy + yα = c/a

                  ∴0×β + βy + y×0 = c/a

                                         βy = c/a

  hence, product of 2 zeros = c/a

Let p(x) =ax³ + bx² + cx + d
Given that, one of the zeroes of the cubic polynomial p(x) is zero.
Let α, β and γ are the zeroes of cubic polynomial p(x), where a = 0.
We know that,

Here p(3) = 0
⇒ 4(3)² - 6 x 3 - m = 0
⇒ 36 - 18 - m = 0 ⇒ m =18
∴ Value of m is exactly divisible by 9.

x - 2 is a factor of x³ - 3x² + 4x - 4
∵ remainder is zero
Similarly x + 1 is a factor of 2x³ + 4x + 6
but x - 1 is not a factor of x⁵ + x⁴ - x³ + x² - x + 1
∵ remainder is not zero
∴ Statements 1 and 2 are correct.