Algebra 2 help me please

Question

Consider the function p(x)+x⁵--3x⁴+3x³-x² whichof the roots of the function have multiplicity of 3

Answer 1

This question would require you to practice your factoring skills :)

x⁵-3x⁴+3x³-x² can first be factored into x²(x³ - 3x² + 3x - 1) by utilizing the GCF method. Next, you may notice that the second factor is the expansion of the binomial (x - 1)³ using the Binomial Theorem.

This leads us to the complete factored form: x⁵-3x⁴+3x³-x² = x²(x - 1)³

Now we can understand the multiplicities of the roots. Multiplicities of roots are determined by the number of times a factor is used in the polynomial. In this case, we can see "x" is used twice as it is raised to the second power. x -1 is used three times as is raised to third power.

If we think of the solutions represented by the factors:

1. x = 0 is a root with multiplicity of 2.
2. x = 1 is a root with multiplicity of 3.