Which of the following statements is true?

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) polynomial of degree $n$ has at most $n$ zeros.For $A$: $p(x) = x^3 + 8$. This is a cubic polynomial (degree $3$),so it has $3$ zeros. Thus,this s

Vedclass · Accepted Answer

$p(x) = x^3 + 8$ has three zeros.

Vedclass · Answer

(A) polynomial of degree $n$ has at most $n$ zeros.For $A$: $p(x) = x^3 + 8$. This is a cubic polynomial (degree $3$),so it has $3$ zeros. Thus,this statement is true.For $B$: $p(x) = x^2 - 36$. This is a quadratic polynomial (degree $2$),so it has at most $2$ zeros. This statement is false.For $C$: $p(x) = 4x^3 + x = x(4x^2 + 1)$. The zeros are $x = 0$ and $4x^2 = -1$ (no real zeros). It has only $1$ real zero,though $3$ complex zeros. However,in the context of standard school mathematics,we often look for real roots. Even if we consider complex roots,$A$ is a more direct example of a cubic polynomial having $3$ roots.For $D$: $p(x) = 4x^3 - x = x(4x^2 - 1) = x(2x - 1)(2x + 1)$. The zeros are $x = 0, 1/2, -1/2$. This also has $3$ zeros.Note: Both $A, C,$ and $D$ have $3$ zeros. However,$p(x) = x^3 + 8$ is a standard cubic form. Given the options,$A$ is a correct statement.

Similar Questions

Obtain a quadratic polynomial with the following conditions:
The sum of the zeros $= -\frac{1}{4}$;
The product of the zeros $= \frac{1}{4}$.

If the polynomial $p(x)$ is divided by $x^{2}+4x+2$, then the quotient polynomial $x^{2}-4x+14$ and the remainder polynomial $9x-13$ are obtained. Find the polynomial $p(x)$.

Find the zeroes of the following polynomial by the factorisation method and verify the relationship between the zeroes and the coefficients of the polynomial:
$t^{3}-2 t^{2}-15 t$

The zeros of the cubic polynomial $p(x) = ax^3 + bx^2 + cx + d$ where $a \neq 0$ and $a, b, c, d \in R$ are $\alpha, \beta,$ and $\gamma$. Then $\alpha \beta \gamma = \dots$

Answer the following and justify: What will be the quotient and remainder on division of $ax^{2} + bx + c$ by $px^{3} + qx^{2} + nx + s$,where $p \neq 0$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module