Faded examples · Using the factor theorem to test a factor
Non-calculator
Each example shows a little less than the one before - complete the faded (blank) steps yourself, using the same three steps every time: read the value to substitute, substitute it, then state the conclusion. Also answer the check question. No calculator.
①Example 1fully worked: read it through
Is $(x-3)$ a factor of $f(x)=x^{3}-4x^{2}+3x$?
1Read off the value to substitute (watch the sign)
$(x-3)$ means substitute $x=3$
2Substitute and evaluate
$f(3)=27-36+9=0$
3State whether it is a factor
yes, it is a factor
Check · Why substitute $3$ and not $-3$?
A $(x-3)$ is zero when $x=3$B because $3$ is positiveC you should use $-3$D either works
②Example 2you finish the last 1 step
Is $(x+2)$ a factor of $f(x)=x^{3}+2x^{2}-x-2$?
1Read off the value to substitute (watch the sign)
$(x+2)$ means substitute $x=-2$
2Substitute and evaluate
$f(-2)=-8+8+2-2=0$
3State whether it is a factor
yes, it is a factor
Check · What would a non-zero answer have told you?
A it is not a factor, and the value is the remainderB you made an errorC it is still a factorD nothing
③Example 3you finish the last 2 steps
Is $(x-1)$ a factor of $f(x)=x^{3}+2x^{2}+3$?
1Read off the value to substitute (watch the sign)
$(x-1)$ means substitute $x=1$
2Substitute and evaluate
$f(1)=1+2+3=6$
3State whether it is a factor
no - the remainder is $6$, so it is not a factor
Check · What is the remainder when $f(x)$ is divided by $(x-1)$?
A $0$B $6$C $1$D $3$
④Example 4your turn: every step
$(x-2)$ is a factor of $x^{3}+kx^{2}-4$. Find $k$.
1Read off the value to substitute (watch the sign)
$(x-2)$ means $f(2)=0$
2Substitute and evaluate
$8+4k-4=0$, so $4k=-4$
3State whether it is a factor
$k=-1$
Check · Why can you set the expression equal to zero?
A a factor means the polynomial vanishes at that valueB because $k$ is unknownC you cannotD because $-4$ is negative
Answers · Factor Theorem
Faded examples · Using the factor theorem to test a factor
① Example 1$(x-3)$ means substitute $x=3$→$f(3)=27-36+9=0$→yes, it is a factor$yes$
Check: A: $(x-3)$ is zero when $x=3$
② Example 2$(x+2)$ means substitute $x=-2$→$f(-2)=-8+8+2-2=0$→yes, it is a factor$yes$
Check: A: it is not a factor, and the value is the remainder
③ Example 3$(x-1)$ means substitute $x=1$→$f(1)=1+2+3=6$→no - the remainder is $6$, so it is not a factor$no$
Check: B: $6$
④ Example 4$(x-2)$ means $f(2)=0$→$8+4k-4=0$, so $4k=-4$→$k=-1$$k=-1$
Check: A: a factor means the polynomial vanishes at that value
Factor Theorem