Polynomial Word Problem

by Mary Clark
(Springrove, Pa)

I have an Algebra Word Problem that I cannot solve. Can you please help. The problem is:

The sum of two integers is 10. If their product is 24,what are the two integers?

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Mar 01, 2012
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starstarstarstarstar 		ANSWER TO MARY CLAECK
by: CEDRUS

A quaratic trinome can be written as follows:

a(x-x')(x-x")=0. Let's develop:

a(x^2 - xx" - xx' + x'x")=0. Let's divide by "a":

x^2 - xx" - xx' + x'x")=0 OR

x^2 - x(x'+x') + x'x" = 0

S(um) = x'+x" and P(roduct) = x'x"

hence x^2 +10x +24 =0

extract the roots.

x'= 6 ad x" = 4
Aug 08, 2011
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starstarstarstarstar 		IMBA
by: Anonymous

no comment
Jan 05, 2010
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starstarstarstarstar 		Polynomial Word Problem Solution
by: Karin

Hi Mary,

Word Problems are really tough for a lot of students. I hope that you are studying Polynomials at this point.

This problem really combines a lot of Algebra skills that hopefully you've already studied. In order to begin this problem, I would write a system of equations. A system of equations means that you would write two equations that represent this problem. In this case, one equation would represent the sum and the other equation would represent the product.

So, if the sum of two integers is 10, the equation is:
x + y = 10 (Since we don't know what the integers are, one will be assigned the variable x and the other, y)

If the product is 24, the equation is:
xy = 24 (product means to multiply)

Now you have two equations:
x + y = 10
xy = 24

From here, I would use the substitution method to solve the system. I would rewrite the first equation as y = -x + 10 (Subtract x from both sides).

Now substitute this equation into xy = 24.

y = -x + 10
xy = 24

x(-x + 10) = 24 Substitute -x + 10 for y.

-x2 + 10x = 24 Distribute the x.

-x2 + 10x - 24 = 24 -24 Set the equation = to 0 so you can factor.

-x2 + 10x -24 = 0

-1[-x2 + 10x -24 = 0] Make the lead coefficient positive by multiplying by -1.

x2 - 10x + 24 = 0.

Now factor this polynomial:

(x - 6) (x - 4)

x = 6 x = 4

The two integers are 6, and 4.

Check:
6 + 4 = 10
6(4) = 24

Mary, I hope this helps. You really need to have a strong understanding of systems of equations and polynomials to solve this problem. Hopefully you do!

Let me know if this helps you!

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