(a) ... passes through the x-axis at x=1 and x=−1?
(x)(x-1)(x+1)
= x(x^2-1)
= x^3-x
y = x^3 - x
(b) ... passes through the origin and touches the x-axis at x=−3?
(x)(x+3)^2 // The reason why x has to be used and not added to anything is to avoid another non-x term that the +3 could be multiplied by to give a non-zero y-intercept. Squaring the (x+3) ensures the curve touches and doesn't cross the x-axis when x=-3.
= x(x^2+6x+9)
y = x^3 + 6x^2 + 9x
(c) ... touches the x-axis at x=2 and crosses the y-axis at 12?
(x-2)^2(x+3) // The 3 multiplies with the (-2)^2 to give a y-intercept of 12
= (x^2-4x+4)(x+3)
= x^3-4x^2+4x+3x^2-12x+12
= x^3-x^2-8x+12
y = x^3 - x^2 - 8x + 12
(d) ... crosses the y-axis at −6 and has three integer roots?
Three integers that multiply to get -6 are -1, -2 and -3.
(x-1)(x-2)(x-3)
= (x-3)(x^2-3x+2)
= x^3-3x^2+2x-3x^2+9x-6
= x^3-6x^2+11x-6
y = x^3 - 6x^2 + 11x - 6
(e) ... crosses the y-axis at y=5 and touches the x-axis at x=1?
(x+5)(x-1)^2 // (x-1)^2 makes it touch at x=1, and the 5 is multiplied by (-1)^2 to give the 5 we want for the y-intercept.
= (x+5)(x^2-2x+1)
= x^3-2x^2+x+5x^2-10x+5
= x^3 + 3x^2 - 9x + 5
The first two curves are not unique. Their initial bracket that contains just an 'x' can be multipled by any non-zero real number to give an answer that fits the description, but is a vertically stretched/compressed version of the curve I gave. However, you are unable to 'stretch' the last three curves, since their y-coordinate has to be fixed.
However, despite (c) and (e) being unique, d is not. I started with three sets of brackets: (x+a)(x+b)(x+c). Any combinations of integers a, b and c such that abc=-6 would be a valid solution. For example, y=(x+1)(x+2)(x-3), which expands to y = x^3 - 7x - 6, is another solution.