Factor Theorem Flashcards

All 7 cards in this deck

State the factor theorem.

If $f(a)=0$, then $(x-a)$ is a factor of the polynomial $f(x)$ (and if $(x-a)$ is a factor, then $f(a)=0$).

Which value of $x$ do you substitute to test whether $2x-3$ is a factor of a polynomial $f(x)$?

$x=\frac{3}{2}$ (solve $2x-3=0$). If $f\left(\frac{3}{2}\right)=0$, then $2x-3$ is a factor.

What are the steps to show a linear expression is a factor of a polynomial?

e.g. show $x-2$ is a factor of $f(x)=x^5-32$

1. Set the linear expression to zero: $x-2=0$, so $x=2$.
2. Substitute: $f(2)=2^5-32=0$.
3. Since $f(2)=0$, $x-2$ is a factor.

True or false? If $f(3)=0$ then $(x+3)$ is a factor of $f(x)$.

False. $f(3)=0$ means $(x-3)$ is a factor.

True or false? The factor theorem can only be used with whole-number values of $x$.

False. It works for rational values too, e.g. test $x=\frac{2}{3}$ to check whether $3x-2$ is a factor.

A cubic is written as $f(x)=(x-1)(2x+1)(x-4)$. What are the roots of $f(x)=0$?

$x=1$, $x=-\frac{1}{2}$, $x=4$ — set each factor equal to zero.

When hunting for a factor of a cubic whose leading coefficient is not $1$, which values of $x$ should you test?

e.g. $f(x)=2x^3-x^2-8x+4$

$x=\frac{p}{q}$, where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient.

e.g. test $\pm1,\pm2,\pm4,\pm\frac{1}{2}$.