Every factor of P (x) is of the form (x-c) where c is a real number is it true or false?

The statement is **true**.

If \( P(x) \) is a polynomial with real coefficients, and every factor of \( P(x) \) is of the form \( (x - c) \), where \( c \) is a real number, then \( P(x) \) can be expressed as:

\[
P(x) = a(x - c_1)(x - c_2)\dots(x - c_n)
\]

Here, \( a \) is a real constant, and \( c_1, c_2, \dots, c_n \) are the real roots (or zeros) of the polynomial. This implies that \( P(x) \) has only real roots, and each factor \( (x - c_i) \) corresponds to a root \( c_i \) of the polynomial.

Would you like more details or have any questions? 

Here are 5 related questions for further exploration:
1. What happens if \( P(x) \) has complex roots?
2. Can a polynomial with real coefficients have non-real factors?
3. How can you determine the number of roots of \( P(x) \)?
4. What is the relationship between the degree of a polynomial and the number of its roots?
5. How does the Fundamental Theorem of Algebra apply to this scenario?

**Tip:** When factoring polynomials, always check if the coefficients and roots are real or complex to understand the nature of the factors.

Explore the truth about polynomials whose factors are of the form (x-c), where c is a real number. This detailed explanation covers how polynomials with real coefficients are structured, emphasizing the role of real roots and factorization. Dive into the mathematical underpinnings suitable for advanced high school students.

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