Question

Zeros of polynomials (with factoring

We want to find the zeros of this polynomial:
p(x) = 2x3 + 5x2 – 2x – 5
Plot all the zeros (x-intercepts) of the polynomial in the interactive graph.

Answer 1

The zeroes are (-1,0), (1, 0) and (-5/2, 0)

Explanation:

We can find the zeroes by factoring:

2x^3 + 5x^2 - 2x - 5 = 0

x^2(2x + 5) - 1(2x + 5) = 0

(x^2 - 1)(2x + 5) = 0

(x - 1)(x + 1)(2x + 5) = 0

So x = -1, 1, -5/2.

Answer 2

x = -5/2 x=1 x = -1

Explanation:

p(x) = 2x^3 + 5x^2 – 2x – 5

Use factor by grouping

p(x) = 2x^3 + 5x^2 – 2x – 5

Factor x^2 from the first group and -1 from the second group

x^2(2x +5) -1( 2x+5)

Then factor out 2x+5

p(x) = (2x+5) (x^2-1)

Factor x^2 -1 as the difference of squares

p(x) = (2x+5)(x-1)(x+1)

Set to zero to find the x intercepts

0 = (2x+5)(x-1)(x+1)

Using the zero product property

2x+5 =0 x-1 =0 x+1 =0

2x = -5 x=1 x=-1

x = -5/2 x=1 x = -1