The zeros of the polynomial are -3, -2, 4 and 6 and the graph of the polynomial is graph (a)
How to factor the polynomial?
The polynomial is given as:
P(x) = x^4 - 5x^3 - 20x^2 + 60x + 144
Expand the polynomial function
P(x) = x^4 - 10x^3 + 5x^3 + 24x^2 - 50x^2 + 6x^2 + 120x - 60x + 144
Rewrite the function as:
P(x) = x^4 - 10x^3 + 24x^2 + 5x^3 - 50x^2 + 120x + 6x^2 - 60x + 144
Factorize the function
P(x) = x^2(x^2 - 10x + 24) + 5x(x^2 - 10x + 24) + 6(x^2 - 10x + 24)
Factor out x^2 - 10x + 24
P(x) = (x^2 + 5x + 6)(x^2 - 10x + 24)
Expand each bracket
P(x) = (x^2 + 3x + 2x + 6)(x^2 - 4x - 6x + 24)
Factorize each bracket
P(x) = [x(x + 3) + 2(x + 3)][x(x - 6) - 4(x - 6)]
Factor out x + 3 and x - 6
P(x) = (x + 3)(x + 2)(x - 4)(x - 6)
Set to 0
(x + 3)(x + 2)(x - 4)(x - 6) = 0
Solve for x
x =-3, -2, 4 and 6
Hence, the zeros of the polynomial are -3, -2, 4 and 6 and the graph of the polynomial is graph (a)
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