Express f(x) as a product of linear factors.
The Correct Answer and Explanation is:
To express a function ( f(x) ) as a product of linear factors, we are essentially factoring the polynomial expression of ( f(x) ) into terms of the form ( (x – r) ), where ( r ) is a root or zero of the polynomial. This is typically done by finding the roots of the equation ( f(x) = 0 ), and then using those roots to write the factored form of the function.
Steps to Express ( f(x) ) as a Product of Linear Factors:
- Find the roots of the polynomial:
- To express ( f(x) ) as a product of linear factors, we first need to find the values of ( x ) that make ( f(x) = 0 ). These are the roots of the polynomial.
- Depending on the degree and form of the polynomial, you can find the roots using different methods, such as factoring, synthetic division, the Rational Root Theorem, or numerical methods (like the quadratic formula for second-degree polynomials).
- Write the linear factors:
- Once the roots (also called zeros) of ( f(x) ) are found, you can express ( f(x) ) as a product of linear factors. If ( r_1, r_2, r_3, \dots, r_n ) are the distinct roots, the polynomial can be factored as:
[
f(x) = a(x – r_1)(x – r_2)(x – r_3) \dots (x – r_n)
]
where ( a ) is the leading coefficient of the polynomial, if necessary.
- Check for multiplicities:
- If any roots have multiplicities greater than one (i.e., the same root appears more than once), the corresponding linear factor will be repeated. For example, if a root ( r ) appears twice, the factor would be ( (x – r)^2 ).
- Verify by expanding:
- After factoring, it’s good practice to expand the product of linear factors to verify that it equals the original polynomial expression.
Example
Consider ( f(x) = x^2 – 5x + 6 ). To express this as a product of linear factors:
- Find the roots:
- We solve ( x^2 – 5x + 6 = 0 ).
- Factor the quadratic: ( (x – 2)(x – 3) = 0 ).
- The roots are ( x = 2 ) and ( x = 3 ).
- Write the linear factors:
- The factored form of ( f(x) ) is ( (x – 2)(x – 3) ).
Thus, ( f(x) = (x – 2)(x – 3) ) is the product of linear factors.
Conclusion
Factoring polynomials into linear factors is a critical algebraic skill. By finding the roots of the polynomial and expressing ( f(x) ) as a product of factors like ( (x – r) ), you can represent any polynomial in its simplest factored form.