The quadratic polynomial

Full Answer Section

Now, solve for $k$: $k=-4/3$

Let's consider the case if the polynomial was meant to be kx2+4x+3. In this case, $a=k$, $b=4$, and $c=3$. Then, $-4/k=3/k$. $-4=3$, which is false. So this is not the correct interpretation.

Let's consider another possibility, if the polynomial was meant to be kx2+4x+3K (capital K for constant term). If the original polynomial was kx2+4x+3K (with $K$ representing a constant term not related to the leading coefficient $k$), and assuming $K$ is the same $k$ as the leading coefficient, then our initial assumption stands.

Given the common format for quadratic polynomials in such problems, the most probable interpretation is that the polynomial is kx2+4x+3k.

Final Answer: The value of $k$ is $-4/3$.

Sample Answer

A quadratic polynomial is generally expressed in the form ax2+bx+c. The given polynomial is $kx+4x+3k$. To correctly identify the coefficients $a$, $b$, and $c$, we need to ensure the polynomial is written in the standard form. It appears there might be a typo in the polynomial, as $kx$ and $4x$ are both terms with $x$.

Assuming the quadratic polynomial is kx2+4x+3k:

Here, $a=k$, $b=4$, and $c=3k$.

For a quadratic polynomial ax2+bx+c: The sum of the zeros is given by $\alpha +\beta =-b/a$. The product of the zeros is given by $\alpha \beta =c/a$.

Given that the sum of the zeros is equal to their product: $-b/a=c/a$

Substitute the values of $a$, $b$, and $c$ from our polynomial: $-(4)/k=(3k)/k$

As long as k=0, we can simplify both sides by multiplying by $k$: $-4=3k$