Polynomials & Factor Theorem Flashcards

All 13 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.

What are the steps to solve a cubic equation by factorising?

e.g. solve $x^3+x^2-10x+8=0$

1. Find a root: $f(1)=1+1-10+8=0$, so $(x-1)$ is a factor.
2. Factorise fully: $(x-1)(x-2)(x+4)=0$.
3. Set each factor to zero: $x=1$, $x=2$, $x=-4$.

What are the steps to find the remaining factors of a cubic when one factor is given?

e.g. $(x+1)$ is a factor of $x^3+2x^2-x-2$

1. Divide the cubic by the given factor (or compare coefficients): $x^2+x-2$.
2. Factorise the quadratic: $(x+2)(x-1)$.
3. Remaining factors: $(x+2)$ and $(x-1)$.

What are the steps to find an unknown coefficient when a linear expression is a factor?

e.g. $(x-2)$ is a factor of $f(x)=x^3+kx^2-4x+4$

1. The factor gives $x=2$, so $f(2)=0$.
2. Substitute: $8+4k-8+4=0$.
3. Solve: $4k+4=0$, so $k=-1$.

What are the steps to solve a cubic exactly when its quadratic factor does not factorise?

e.g. solve $(x-2)(x^2-2x-2)=0$

1. First factor gives $x=2$.
2. Use the quadratic formula on $x^2-2x-2=0$: $x=\frac{2\pm\sqrt{12}}{2}$.
3. Simplify the surd: $x=2$, $x=1+\sqrt{3}$, $x=1-\sqrt{3}$.

What are the steps to form a cubic equation from a volume problem?

e.g. a cuboid with sides $x$, $x+1$ and $x+3$ has volume $24$

1. Multiply the dimensions: $x(x+1)(x+3)$.
2. Expand: $x^3+4x^2+3x$.
3. Set equal to the given volume and rearrange: $x^3+4x^2+3x-24=0$.

What are the steps to fully factorise a cubic with leading coefficient $1$?

e.g. factorise $x^3-2x^2-5x+6$

1. Test $x=$ factors of the constant term until $f(x)=0$: $f(1)=0$, so $(x-1)$ is a factor.
2. Divide (or compare coefficients) to get the quadratic: $x^2-x-6$.
3. Factorise the quadratic: $(x-1)(x-3)(x+2)$.

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