Mathematics 9709/23 — May/June 2019

Walkthrough

We are told that $(x + 1)$ is a factor of $p(x)$. By the Factor Theorem, this means that $x = -1$ is a root of $p(x)$, so $p(-1) = 0$. The most direct way to use this is to substitute $x = -1$ into the polynomial.

When we substitute, we need to be careful with signs:

• $(-1)^3 = -1$, so $4(-1)^3 = -4$.
• $(-1)^2 = 1$, so $(k+1)(-1)^2 = k+1$.
• $-m(-1) = +m$.
• The final term is $+3k$.

So the equation becomes:

$-4 + (k+1) + m + 3k = 0$

Now combine the constant terms and the $k$ terms:

$-4 + 1 = -3$

and

$k + 3k = 4k$

So we have:

$-3 + 4k + m = 0$

Finally, rearrange to make $m$ the subject:

$m = 3 - 4k$

This is the required expression for $m$ in terms of $k$.

Key Takeaways

The key idea is the Factor Theorem: if $(x - a)$ is a factor of a polynomial $p(x)$, then $p(a) = 0$. Here the factor is $(x + 1)$, which is the same as $(x - (-1))$, so we substitute $x = -1$. This turns the polynomial equation into an equation involving the unknown constants, which can then be solved.

Common Mistakes

A common mistake is mishandling the sign of $-m(-1)$: it becomes $+m$, not $-m$. Another common mistake is forgetting that $(k+1)(-1)^2 = k+1$, since $(-1)^2 = 1$. Also, students sometimes forget to set the whole expression equal to zero. The mark scheme allows algebraic long division or an identity with the remainder equated to zero, but the method must be shown; an unsupported answer is not sufficient.

Things to Be Careful About

Be careful when substituting negative values into powers: $(-1)^3 = -1$ but $(-1)^2 = 1$. When combining terms, keep the $k$ terms separate from the constant terms: $k + 3k = 4k$ and $-4 + 1 = -3$. Finally, make sure the final rearrangement is correct: starting from $-3 + 4k + m = 0$, adding $3$ and subtracting $4k$ from both sides gives $m = 3 - 4k$.