+1904 To find an equation that has the three solutions of 2, 4, and 5, first set each solution equal to x.
\(x=2\)
\(x=4\)
\(x=5\)
Next, figure out what number goes with each of the three solutions that makes x equal to zero on the right side. For x=2, the number will be -2; for x=4, the number is -4; for x=5, the number will be -5. What you do on the right side, you do to the left side as well.
\(x=2\)
\(x-2=2-2\)
\(x-2=0\)
\(x=4\)
\(x-4=4-4\)
\(x-4=0\)
\(x=5\)
\(x-5=5-5\)
\(x-5=0\)
Next, combine all three equations using the zero product.
\((x-2)\times(x-4)\times(x-5)=0\)
Next, expand the left side.
\((x-2)\times(x-4)\times(x-5)=0\)
\(({x}^{2}-4x-2x+8)\times(x-5)=0\)
\(({x}^{2}-6x+8)\times(x-5)=0\)
\({x}^{3}-5{x}^{2}-6{x}^{2}+30x+8x-40=0\)
\({x}^{3}-11{x}^{2}+38x-40=0\)
Next, replace 0 on the right side with y and then switch the equation around so that the y is on the left side.
\({x}^{3}-11{x}^{2}+38x-40=y\)
\(y={x}^{3}-11{x}^{2}+38x-40\)
+1904 To find an equation that has the three solutions of 2, 4, and 5, first set each solution equal to x.
\(x=2\)
\(x=4\)
\(x=5\)
Next, figure out what number goes with each of the three solutions that makes x equal to zero on the right side. For x=2, the number will be -2; for x=4, the number is -4; for x=5, the number will be -5. What you do on the right side, you do to the left side as well.
\(x=2\)
\(x-2=2-2\)
\(x-2=0\)
\(x=4\)
\(x-4=4-4\)
\(x-4=0\)
\(x=5\)
\(x-5=5-5\)
\(x-5=0\)
Next, combine all three equations using the zero product.
\((x-2)\times(x-4)\times(x-5)=0\)
Next, expand the left side.
\((x-2)\times(x-4)\times(x-5)=0\)
\(({x}^{2}-4x-2x+8)\times(x-5)=0\)
\(({x}^{2}-6x+8)\times(x-5)=0\)
\({x}^{3}-5{x}^{2}-6{x}^{2}+30x+8x-40=0\)
\({x}^{3}-11{x}^{2}+38x-40=0\)
Next, replace 0 on the right side with y and then switch the equation around so that the y is on the left side.
\({x}^{3}-11{x}^{2}+38x-40=y\)
\(y={x}^{3}-11{x}^{2}+38x-40\)
