how can i find the zeros

Question

Factor the polynomial and use the factored form to find the zeros. (Enter your answers as a comma-separated list. Enter all answers using the appropriate multiplicities.)

P(x) = x4 − 5x3 + 4x2

Answer 1

graph it and see where it intersects the x axis

zeros = x intercepts = roots

4 zeros

0,1, and 4, but the curve is tangent to the x axis at x=0 so it's a "double zero" like James Bond is double 0 seven

or

factor

x^4 -5x^3 +4x^2

= x^2(x^2 -5x+4)

= x^2(x-1)(x-4)

= (x-0)(x-0)(x-1)(x-4)

set each factor = 0 and solve for x

roots are 0, 0, 1 and 4= the four zeros

Answer 2

P(x) = x⁴ − 5x³ + 4x²

x² is a common factor of all three terms so we can write

P(x) = x²(x² − 5x + 4)

The quadratic can be factored by finding a pair of numbers

1. whose sum is –5, and
2. whose product is 4

That pair of numbers is –4 and –1, so we have the factored form of our polynomial:

P(x) = x²(x – 4)(x – 1)

So the zeros are

0, 4, 1

The first zero (0) has a multiplicity of 2.

[ See if you can also use the method above to find the zeroes of P(x) = x³ − x² − 12x ]