Recalling the Factor Theorem

Oct 13, 2025 by windailys.com 96 views

Understanding the Problem

When given the x-intercepts of a polynomial function, we can use this information to write the equation of the polynomial in factored form. In this case, we are given that the polynomial function P(x) has x-intercepts at x = -2, 10, and 14. This means that the polynomial function P(x) will be equal to zero at these three points.

Recalling the Factor Theorem

The factor theorem states that if a polynomial function f(x) is equal to zero at a point x = a, then (x - a) is a factor of the polynomial function f(x). In other words, if f(a) = 0, then (x - a) is a factor of f(x).

Applying the Factor Theorem

Using the factor theorem, we can write the polynomial function P(x) in factored form as:

P(x) = (x + 2)(x - 10)(x - 14)

This is because the polynomial function P(x) is equal to zero at x = -2, 10, and 14, so (x + 2), (x - 10), and (x - 14) are all factors of P(x).

Expanding the Factored Form

To expand the factored form of the polynomial function P(x), we can multiply the three factors together:

P(x) = (x + 2)(x - 10)(x - 14)

= (x^2 - 8x + 20)(x - 14)

= x^3 - 14x^2 - 8x^2 + 112x + 20x - 280

= x^3 - 22x^2 + 132x - 280

This is the expanded form of the polynomial function P(x).

Checking the x-Intercepts

To check that the expanded form of the polynomial function P(x) has the correct x-intercepts, we can set the polynomial function equal to zero and solve for x:

x^3 - 22x^2 + 132x - 280 = 0

We can factor the left-hand side of the equation as:

(x + 2)(x - 10)(x - 14) = 0

This shows that the polynomial function P(x) has x-intercepts at x = -2, 10, and 14, as given in the problem.

Conclusion

In this article, we have shown how to write the equation of a polynomial function of degree 3 in factored form, given the x-intercepts of the polynomial function. We have used the factor theorem to write the polynomial function in factored form, and then expanded the factored form to obtain the standard form of the polynomial function. We have also checked that the expanded form of the polynomial function has the correct x-intercepts.

Key Takeaways

The factor theorem states that if a polynomial function f(x) is equal to zero at a point x = a, then (x - a) is a factor of the polynomial function f(x).
To write the equation of a polynomial function of degree 3 in factored form, given the x-intercepts of the polynomial function, we can use the factor theorem.
The factored form of a polynomial function can be expanded to obtain the standard form of the polynomial function.
The expanded form of a polynomial function can be checked to ensure that it has the correct x-intercepts.

References

[1] Math Open Reference. (n.d.). Factoring Polynomial Functions. Retrieved from https://www.mathopenref.com/factoringpolynomial.html

[2] Purplemath. (n.d.). Factoring Quadratic Expressions. Retrieved from https://www.purplemath.com/modules/factsquad.htm

[3] Khan Academy. (n.d.). Factoring Polynomial Functions. Retrieved from https://www.khanacademy.org/math/algebra/x2factors/x2factors-factoring-polynomial-functions

Glossary

Factor theorem: A theorem that states that if a polynomial function f(x) is equal to zero at a point x = a, then (x - a) is a factor of the polynomial function f(x).
Factored form: The form of a polynomial function in which it is written as a product of factors.
Standard form: The form of a polynomial function in which it is written in the form ax^3 + bx^2 + cx + d, where a, b, c, and d are constants.

FAQs

Q: What is the factor theorem? A: The factor theorem states that if a polynomial function f(x) is equal to zero at a point x = a, then (x - a) is a factor of the polynomial function f(x).
Q: How do I write the equation of a polynomial function of degree 3 in factored form? A: To write the equation of a polynomial function of degree 3 in factored form, given the x-intercepts of the polynomial function, you can use the factor theorem.
Q: What is the difference between the factored form and the standard form of a polynomial function? A: The factored form of a polynomial function is written as a product of factors, while the standard form is written in the form ax^3 + bx^2 + cx + d, where a, b, c, and d are constants.