Which approach involves substituting potential zeros until one yields zero, and then applying synthetic division to factor further?

Which approach involves substituting potential zeros until one yields zero, and then applying synthetic division to factor further?

• A. Rational Zero Theorem
• B. Factor Theorem
• C. Polynomial Remainder Theorem
• D. Descartes' Rule of Signs

Answer: (A)

The idea is to combine a targeted search for rational zeros with a division step that reveals more factors. The Rational Zero Theorem tells you which rational numbers could be zeros: they are factors of the constant term divided by factors of the leading coefficient. You test these candidates by substituting them into the polynomial to see if they make it zero. When a candidate makes the polynomial zero, the Factor Theorem guarantees that x minus that number is a factor. You then use synthetic division to divide by (x − r), obtaining a simpler quotient polynomial. That quotient can be factored in the same way, peeling away roots one by one until the whole polynomial is factored. This approach directly links identifying plausible zeros with an efficient division step to factor further.

The idea is to combine a targeted search for rational zeros with a division step that reveals more factors. The Rational Zero Theorem tells you which rational numbers could be zeros: they are factors of the constant term divided by factors of the leading coefficient. You test these candidates by substituting them into the polynomial to see if they make it zero. When a candidate makes the polynomial zero, the Factor Theorem guarantees that x minus that number is a factor. You then use synthetic division to divide by (x − r), obtaining a simpler quotient polynomial. That quotient can be factored in the same way, peeling away roots one by one until the whole polynomial is factored. This approach directly links identifying plausible zeros with an efficient division step to factor further.