Math · College algebra · Worked example

Write a polynomial from its zeros

Find the cubic polynomial with a zero at −1, a zero of multiplicity 2 at 2, and y-intercept −8.

Build the factors

Each zero r contributes a factor x − r raised to its multiplicity: f(x) = a(x + 1)(x − 2)². The constant a is still unknown.

Use the y-intercept

f(0) = −8 fixes a.

\begin{array}{l} a (0 + 1) {(0 - 2)}^{2} = - 8 \\ 4 a = - 8 \\ a = - 2 \end{array}

Expand

The standard form shows degree 3 and leading coefficient −2.

\begin{array}{l} - 2 (x + 1) {(x - 2)}^{2} \\ - 2 x^{3} + 6 x^{2} - 8 \end{array}

Check the shape

With odd degree and a negative leading coefficient, the graph rises on the left and falls on the right. It crosses the axis at −1 and touches it at 2.

Result

f(x) = −2(x + 1)(x − 2)², or −2x³ + 6x² − 8 in standard form.

Your turn

Write the cubic with zeros 0, 3 and −3 and leading coefficient 2.

Show the answer and explanation

f(x) = 2x(x − 3)(x + 3) = 2x³ − 18x.

The zeros give the factors x, x − 3 and x + 3; multiplying by the leading coefficient 2 and expanding gives 2x³ − 18x.

\begin{array}{l} 2 x (x - 3) (x + 3) \\ 2 x^{3} - 18 x \end{array}

Keep exploring

Graph plots f with the zeros and the y-intercept marked.

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Graph the polynomial Check the expansion in Math Open worked example on a board Degree and leading term in Math Reference

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