Show that 2x+3 is a factor of 2x^3+5x^2-37x-60. Find all the factors of the polynomial.
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(2x+3) will be a factor of f(x)=2x³+5x²-37x-60 when x=-3/2
∴, f(-3/2)=2(-3/2)³+5(-3/2)²-37(-3/2)-60
=-27/4+45/4+111/2-60
=(-27+45+222-240)/4
=0
∴, (2x+3) is a factor of f(x).
2x³+5x²-37x-60
=2x³+3x²+2x²+3x-40x-60
=(2x+3)(x²+x-20)
=(2x+3)(x²+5x-4x-20)
=(2x+3){x(x+5)-4(x+5)}
=(2x+3)(x+5)(x-4)
∴, the factors of f(x) are (2x+3),(x+5) and (x-4).
∴, f(-3/2)=2(-3/2)³+5(-3/2)²-37(-3/2)-60
=-27/4+45/4+111/2-60
=(-27+45+222-240)/4
=0
∴, (2x+3) is a factor of f(x).
2x³+5x²-37x-60
=2x³+3x²+2x²+3x-40x-60
=(2x+3)(x²+x-20)
=(2x+3)(x²+5x-4x-20)
=(2x+3){x(x+5)-4(x+5)}
=(2x+3)(x+5)(x-4)
∴, the factors of f(x) are (2x+3),(x+5) and (x-4).
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