Multiplying binomials with proofs

Question

Multiply (x-6)(x+2) and provide a proof for the solution.

I don't know where to start and how to do the proof part, isnt the solution itself the proof?

Answer 1

When given two binomials such as: (x - 6)(x +2)

F.O.I.L or First, Outer, Inner, Last

(x - 6)(x + 2) use distributive property

[x * x] First terms

[x * 2] Outer terms

[- 6 * x] Inner terms

[- 6 * 2] Last terms

x² + 2x + ( - 6x) -12

x² + (2x - 6x) - 12 Combine like terms

x² + (-4x) -12 Simplify

x² - 4x -12 This will be your new expression

Take binomials and find values for x.

(x - 6)

x - 6 = 0

x - 6 + 6 = 0 + 6

x = 6

(x + 2)

x + 2 = 0

x + 2 - 2 = 0 - 2

x = -2

So now we found that x has two values 6 and -2. now we must show the proof. Do this by taking our x values and plugging them into the expression we found after multiplying the binomials. So we will have two equations.

x² - 4x - 12

(6)² - 4(6) - 12

36 - 24 - 12

0

(-2)² - 4(-2) - 12

4 + 8 + 12

0

Thus we have found the expression for the binomial product, found values for x, and proven that both the values and expression are correct.

Answer 2

So you need to multiply binomials “with proof”, using (x-6)(x+2) as a particular case in point. You first need the proof part. Consider the product of 2 binomials, using the Distributive property in Multiplication:

Of course, in practice, you would simply do: (x-6)(x+2) = x² + 2x - 6x - 12 = x² - 4x - 12

Now the Proof part: (ax + b)(cx + d) = (ax)(cx) + (ax)d + (cx)b + bd = (ac)x² + (ad + bc)x + bd

In our case of (x-6)(x+2), we have a = c = 1, b = -6, d = 2 ==>

(x-6)(x+2) = (1)(1)x² + [(1)(2) + (-6)(1)]x + (-6)(2) = x² + (2 - 6)x - 12 = x² - 4x - 12

…which you have already gotten without proof.

Note: (x + a) is not the only kind of binomial in general existence… you really want something general, like the form (ax + b)… OR… alternatively, you may choose to “prove” a specific case of the product of 2 simpler binomials, but you need to say that !

You would preface your proof with this as a condition, by being precise, and say: Given 2 binomials “of the form” (x + a) and (x + b), then (x + a)(x + b) = …<show your work here>... = x² + (a + b)x + ab

To your question: “isn't the solution itself the proof?"

No, the solution is not a proof… well, it is proof for this specific case… but what is meant by the word “proof” here is a general case (or at least, some kind of general case)… one that works for _all_ binomial products (of the form you choose)… your solution is but 1 “application" of your proof.

Answer 3

The key to creating the proof for this problem is to apply the distributive property to (x - 6)(x + 2). With the distributive property, the general rule is a(b + c) = ab + ac. Since (x - 6) contains two terms, both the x and -6 will be distributed into the (x + 2), and all the terms we end up with from executing the distributions will be added together. x(x + 2) gives us x^2 + 2x, and -6(x + 2) gives us -6x - 12. Adding all the terms we ended up with gives us x^2 + 2x - 6x - 12 = x^2 - 4x - 12. x^2 - 4x - 12 ends up being our final solution.

Answer 4

(x-6)(x+2) use FOIL, products of First, Outside, Inside, Last terms

= x^2 +2x -6x -6(2)

= x^2 -4x-12

one proof might be graphically. graph the quadratic, it's a parabola and find it's x intercepts

they should be 6 and -2 from (x-6)(x+2) setting each factor = 0 and solving for x. use a graphing calculator, Desmos or TI83, or plot points and sketch the graph

or

use the quadratic formula

x = 4/2 + or - .5sqr(16+48) = 2 +/-.5sqr64 = 2+/-4 = 6 or -2

Answer 5

To multiply (x-6)(x+2) we first use the distributive property: x(x+2) -6(x+2)

x•x +2•x -6•x -6•2

Then complete multiplication in each term: x² +2x - 6x -12

Finally, combine like terms: x² -4x -12

Answer 6

Probably looking for something like this:

(x+6) (x + 2)

[(x+6)x + (x+6)2] - distributive property for multiplication over addition (distributing a binomial (x+6))

[x(x+6) + 2(x+6)] -commutative property for multiplication

[x² + 6x + 2x + 12] - distributive property multiplication over addition

x² + 8x + 12 - combine similar terms -- because 6x + 2x can be written as (6 + 2)x = 8x

Of course doing all of the above, eventually leads to the memory aid FOIL for multiplying two binomials mentally (and quickly).