How to Form a Polynomial from Given Zeroes?

Creating a polynomial from given zeroes is a straightforward process. Let’s break it down step by step.

Step-by-Step Process

1. Identify the Zeroes
   First, you need to know the zeroes of the polynomial. Zeroes are the values of $x$ for which the polynomial equals zero. For example, let’s say we have zeroes $x = 2$, $x = -3$, and $x = 4$

2. Write the Factors
   For each zero, write a factor in the form
   $(x – text{zero})$. For our example, the factors would be:
   • $(x – 2)$ for $x = 2$
   • $(x + 3)$ for $x = -3$
   • $(x – 4)$ for $x = 4$

3. Form the Polynomial
   Multiply these factors together to form the polynomial. For our example, the polynomial would be:

   $P(x) = (x – 2)(x + 3)(x – 4)$

4. Expand the Polynomial (Optional)
   You can expand the polynomial to get it into standard form. Let’s expand our example:

   $(x – 2)(x + 3)(x – 4)$

   First, multiply the first two factors:

   $(x – 2)(x + 3) = x^2 + 3x – 2x – 6 = x^2 + x – 6$

   Next, multiply the result by the third factor:

   $(x^2 + x – 6)(x – 4)$

   $= x^3 – 4x^2 + x^2 – 4x – 6x + 24$

   $= x^3 – 3x^2 – 10x + 24$

   So, the polynomial in standard form is:

   $P(x) = x^3 – 3x^2 – 10x + 24$

Special Cases

Repeated Zeroes

If a zero is repeated, include it as many times as it repeats. For example, if $x = 2$ is a zero twice, the factor $(x – 2)$ will appear twice:

$P(x) = (x – 2)^2(x + 3)$

Complex Zeroes

If the zeroes are complex, they will appear in conjugate pairs. For example, if $x = 2 + i$ is a zero, then $x = 2 – i$ will also be a zero. The factors will be:

$(x – (2 + i))(x – (2 – i))$

Conclusion

Forming a polynomial from given zeroes involves creating factors from each zero and then multiplying these factors together. This process is fundamental in algebra and helps us understand the relationship between zeroes and polynomials.

Citations

1. 1. Khan Academy – Polynomials
2. 2. Purplemath – Polynomials
3. 3. MathIsFun – Polynomials