All ACT Math Resources
Example Question #7 : Polynomials
Step 1: Use the distributive property
Step 2: Combine like terms
Example Question #8 : Polynomial Operations
Two positive consecutive whole numbers that are even are both multiples of . The product of the two numbers is . What is the sum of the two integers?
The question provides two positive whole numbers that are each multiples of 6 and also 6 numbers apart. This may be translated into variables, where the first number may be represented by "x" and the second number may be represented as "x+6" given that it is 6 numbers greater than x.
The problem provides the information that the product of these two numbers is 72. Using the new definitions for the numbers, this may be represented as:
(x)(x+6) = 72
This provides an equation that multiplies two polynomials (one with one term, which is a monomial and one with two terms, which is a binomial) and an ability to solve for what x (the first of the two numbers) may be.
Using FOIL, the result is . This may be rewritten as , which will provide the value of x after factoring.
, where (-6)(12) will provide the product of -72 and the sum of 12 and -6 will yield 6. The results indicate x as having two possible solutions: x=6 and x=-12. Returning back to the question, the goal is to find two positive numbers. This means that x=-12 is not a viable solution and that x=6 is. Now, revisiting the terms used to redefine the two numbers [x and x+6], x has been calculated. After substituting in the x value for the second term, the second number is 12 (6+6=12).
The final step of the problem is to solve for the sum of these two numbers:
Example Question #8 : Polynomials
Multiply these polynomials out and expand.
The proper way to multiply polynomials is to essentially apply the distributive property continually.
So is really the same as , and so on.
Then, simply group together like terms to get the final answer: