How to find the zeros of polynomials
using factoring, division of polynomials and the rational root theorem

. Grade 12 maths questions are presented along with detailed solutions and graphical interpretations.

\( p(x) \) has a zero at \( x = 2 \) and therefore \( \; x - 2 \) is a factor of \( p(x) \). Divide \( p(x) \) by \( x - 2 \)

\( \dfrac{p(x)}{x - 2} = \dfrac {x^3 + 5 x^2 - 2 x - 24}{x-2} = x^2 + 7 x + 12 \)

Using the division above, \( p(x) \) may now be written in factored form as follows:

\( p(x) = (x - 2)(x^2 + 7 x + 12) \)

Factor the quadratic expression \( x^2 + 7 x + 12 \).

\( p(x) = (x - 2)(x + 3)(x + 4) \)

The zeros are found by solving the equation.

\( p(x) = (x - 2)(x + 3)(x + 4) = 0 \)

For p(x) to be equal to zero, we need to have

\( x - 2 = 0 \), or \( x + 3 = 0 \) , or \( x + 4 = 0 \)

Solve each of the above equations to obtain the zeros of \( p(x) \).

\( x = 2 \), \( x = - 3 \) and \(x = - 4 \).

Zeros at \( x = \pm \sqrt5 \) correspond to the factors.

\( x - \sqrt 5 \) and \( x + \sqrt 5 \)

Hence polynomial \( p(x) \) may be written as

\( p(x) = (x - √5)(x + √5) Q(x) = (x^2 - 5)Q(x) \)

Find \( Q(x) \) using long division of polynomials

\( Q(x) = \dfrac{p(x)}{x^2 - 5} \)

\( \quad = \dfrac{3 x^4 + 5 x^3 - 17 x^2 - 25 x + 10}{x^2 - 5} \)

\( \quad = 3 x^2 + 5 x - 2 \)

Factor \( Q(x) = 3 x^2 + 5 x - 2 \)

\( Q(x) = 3 x^2 + 5 x - 2 = (3x - 1)(x + 2) \)

Factor \( p(x) \) completely

\( p(x) = (x - \sqrt 5)(x + \sqrt 5)(3x - 1)(x + 2) \)

Set each of the factors of p(x) to zero to find the zeros.

\( x = \pm \sqrt 5 , x = \dfrac{1}{3} , x = - 2 \)

a) Show that \( x = 1 \) is a zero of multiplicity \( 2 \).

b) Find all zeros of \( p \).

c) Sketch a possible graph for \( p \).

a) If \( x = 1 \) is a zero of multiplicity \( 2 \), then \( (x - 1)^2 \) is a factor of \( p(x) \) and a division of \( p(x) \) by \( (x - 1)^2 \) must give a remainder equal to \( 0 \). A long division gives

\( \dfrac{p(x)}{(x - 1)^2} = \dfrac{x^4 - 2x^3 - 2x^2 + 6x - 3}{(x - 1)^2} = x^2 - 3 \)

The remainder in the division of \( p(x) \) by \( (x - 1)^2 \) is equal to \( 0 \) and therefore \( x = 1 \) is a zero of multiplicity \( 2 \).

b) Using the division above, p(x) may now be written in factored form as follows

\( p(x) = (x - 1)^2(x^2 - 3) \)

Factor the quadratic expression \( x^2 - 3.

\( p(x) = (x - 1)^2 (x - \sqrt 3) (x + \sqrt 3) \)

The zeros are found by solving the equation.

\( p(x) = (x - 1)^2 (x - \sqrt 3) (x + \sqrt 3) = 0 \)

For \( p(x) \) to be equal to zero, we need to have

\( (x - 1)^2 = 0 \) , or \( x - \sqrt 3 = 0 \) , or \( x + \sqrt 3 = 0 \)

Solve each of the above equations to obtain the zeros of p(x).

\( x = 1 \) (multiplicity \( 2 \) ) , \( x = \sqrt 3 \) and \( x = - \sqrt 3 \)

c) With the help of the factored form of \( p(x) \) and its zeros found above, we now make a table of signs.

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We use the zeros of p(x) which graphically are shown as x intercepts, the table of signs and the y intercept (0 , -3) to complete the graph as shown below.

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Rational Zeros Theorem: If \( p(x) \) is a polynomial with integer coefficients and if \( \dfrac{m}{n} \) (in lower terms) is a zero of \( p(x) \), then \( m \) is a factor of the constant term \( 2 \) of \( p(x) \) and n is a factor of the leading \( 6 \) coefficient of \( p(x) \).

Find factors of \( 2 \) and \( 6 \).

factors of \( 2 \) are : \( \quad \pm 1 \) , \( \pm 2 \)

factors of \( 6 \) are: \( \quad \pm 1 , \pm 2 , \pm 3 , \pm 6 \)

possible zeros: divide factors of \( 2 \) by factors of \( 6 \): \( \quad \pm 1 , \pm \dfrac{1}{2} , \pm \dfrac{1}{3} , \pm \dfrac{1}{6} , \pm 2 , \pm \dfrac{2}{3} \)

Because of the large list of possible zeros, we graph the polynomial and guess the zeros from the location of the \( x \) intercepts. Below is the graph of the the given polynomial \( p(x) \) and we can easily see that the zeros are close to \( - \dfrac{1}{3} \), \( \dfrac{1}{2} \) and \( 2 \).

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We now calculate \( p(-\dfrac{1}{3}), p(\dfrac{1}{2}) \) and \( p(2) \) to finally check if these are the exact zeros of \( p(x) \).

\( p \left(-\dfrac{1}{3} \right) = 6\left(-\dfrac{1}{3} \right)^3 - 13 \left (-\dfrac{1}{3} \right)^2 + \left (-\dfrac{1}{3} \right) + 2 = 0 \)

\( p \left(\dfrac{1}{2} \right) = 6 \left (\dfrac{1}{2} \right)^3 - 13 \left (\dfrac{1}{2} \right)^2 + \left (\dfrac{1}{2} \right) + 2 = 0 \)

\( p(2) = 6(2)^3 - 13(2)^2 + (2) + 2 = 0 \)

We have used the rational zeros theorem and the graph of the given polynomial to determine the \( 3 \) zeros of the given polynomial which are \( -\dfrac{1}{3}, \dfrac{1}{2} \) and \( 2 \).

Factor Polynomials

Long Division of Polynomials

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Factor Polynomials by Grouping

Find Zeros of Polynomial Functions

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