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Cubic roots and coefficients

The roots in polynomials are closely linked to the coefficients of that polynomial.

All polynomials in this chapter will be written as follows - the letters are important here:

Sum of roots

Section titled “Sum of roots”

If , and are roots of , then .

This means that the sum of roots of a cubic is equal to .

Find the sum of the roots of

Section titled “Find the sum of the roots of ”
  • The sum of roots is :
  • Answer: sum of roots =

Find the sum of the roots of

Section titled “Find the sum of the roots of ”
  • The sum of roots is :
  • Answer: sum of roots =

The sum of the roots of the polynomial is 2. Find the value of .

Section titled “The sum of the roots of the polynomial is 2. Find the value of .”
  • Sum of roots = which represents .
  • Answer:

The sum of the roots of the polynomial is -4. Find the value of .

Section titled “The sum of the roots of the polynomial is -4. Find the value of .”
  • Sum of roots = which represents .
  • Answer:

Sum of products of roots taken two at a time (sum of product ‘pairs’)

Section titled “Sum of products of roots taken two at a time (sum of product ‘pairs’)”

If , and are roots of , then .

In other words, if we multiply together each possible pair of roots and then add those products, we get the same result as .

Find the sum of product pairs for

Section titled “Find the sum of product pairs for ”
  • The sum of product pairs is :
  • Answer: sum of product pairs =

Find the sum of product pairs for

Section titled “Find the sum of product pairs for ”
  • The sum of product pairs is :
  • Answer: sum of product pairs =

The actual roots of this polynomial (+ some others here) are complex numbers. This shows us that actually, the answer above isn’t really an answer! If we found the actual roots and found the sum of their product pairs, we wouldn’t get a real number.

Product of roots

Section titled “Product of roots”

If , and are roots of , then .

In words, the product of roots of a cubic is equal to .

Find the product of the roots of

Section titled “Find the product of the roots of ”
  • The product of roots is :
  • Answer: product of roots =

Find the product of the roots of

Section titled “Find the product of the roots of ”
  • The product of roots is :
  • Answer: product of roots =

The equation has product of roots . Find the value of .

Section titled “The equation has product of roots . Find the value of .”
  • The product of roots is , which represents .
  • Answer:

The equation has roots , and . Find the value of .

Section titled “The equation has roots , and . Find the value of .”
  • Sum of product pairs = :
    • So .
  • Product of roots = :
    • So .
  • Therefore,
  • Answer:

Summary of finding roots and coefficients

Section titled “Summary of finding roots and coefficients”
Coefficient linkPositive/Negative
Sum of rootsNegative
Sum of products pairsPositive
Product of rootsNegative