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Question: How do you FOIL\[(x - 1)(x + 4)(x - 3)\]?...

How do you FOIL(x−1)(x+4)(x−3)(x - 1)(x + 4)(x - 3)?

Explanation

Solution

FOIL is nothing but a method to multiply polynomials, FOIL (First, Outer, Inner, and Last) the sum of all these terms. Which will give the resultant expression that we want. Solve the given equation by considering the first two parameters to the second using FOIL. Then also apply the same to the resultant and third parameters to get the solution.

Formula Used: FOIL has a formula if the expression is in the form of above diagram (a+b)(c+d)(a + b)(c + d)
First: a×ca \times c
Outer: a×da \times d
Inner: b×cb \times c
Last: b×db \times d
The sum will be ac+ad+bc+bdac + ad + bc + bd

Complete step-by-step solution:
Before solving consider the above diagram in which we will use FOIL to solve the expression. In the same way we will solve this equation.
(x−1)(x+4)(x−3)(x - 1)(x + 4)(x - 3)
First let us consider the first two parameters and then apply FOIL.
(x−1)(x+4)(x - 1)(x + 4)
Sum of First, Outer, Inner, Last
First: (x)x=x2(x)x = {x^2}
Outer (Outside): (x)×4=4x(x) \times 4 = 4x
Inner: (−1)×x=−x( - 1) \times x = - x
Last:- (−1)×4=−4( - 1) \times 4 = - 4
Sum of all these terms will be (x2+4x−x−4)=(x2+3x−4)({x^2} + 4x - x - 4) = ({x^2} + 3x - 4)
Now we will apply FOIL for the expression (x2+3x−4)(x−3)({x^2} + 3x - 4)(x - 3)
Either consider x2+3x{x^2} + 3xas a or 3x−43x - 4 as, I am considering first one and proceeding
We will have ((x2+3x)−4)(x−3)(({x^2} + 3x) - 4)(x - 3)applying FOIL again will be
First: (x2+3x)(x)=x3+3x2({x^2} + 3x)(x) = {x^3} + 3{x^2}
Outer (Outside): (x2+3x)(−3)=−3x2−9x({x^2} + 3x)( - 3) = - 3{x^2} - 9x
Inner: (−4)x=−4x( - 4)x = - 4x
Last: (−4)(−3)=12( - 4)( - 3) = 12
Sum of the terms will be

⇒x3+3x2−3x2−9x−4x+12 ⇒x3−9x−4x+12 ⇒x3−13x+12  \Rightarrow {x^3} + 3{x^2} - 3{x^2} - 9x - 4x + 12 \\\ \Rightarrow {x^3} - 9x - 4x + 12 \\\ \Rightarrow {x^3} - 13x + 12 \\\

The Solution for the FOIL (x−1)(x+4)(x−3)(x - 1)(x + 4)(x - 3)is x3−13x+12{x^3} - 13x + 12

Additional information: FOIL is a method to multiply the polynomial terms in the parameters which is a standard to multiply. You can even multiply with the first term with the other terms in parameters and second term with another two terms in another parameter. The sum will be the same either in both ways. (a+b)(c+d)(a + b)(c + d)

a(c+d)+b(c+d) a×c+a×d  a(c + d) + b(c + d) \\\ a \times c + a \times d \\\

Then
b×c+b×db \times c + b \times d
The sum will be same ac+ad+bc+bdac + ad + bc + bd

Note: When the terms get bigger in one parameter, you can combine the terms and consider them as one to apply FOIL. Here in the below example when three terms are there we will consider the first two as one and apply FOIL.

=>(x+y+z) =>((x+y)+z)  = > (x + y + z) \\\ = > ((x + y) + z) \\\