Polynomials - ACT Math

In adding to both sides:

. . .and adding to both sides:

. . .the variables are isolated to become:

After dividing both sides by , the equation becomes:

The degree of a polynomial is determined by the term with the highest degree. In this case that is , which has a degree of .

Line up each expression vertically. Then combine like terms to solve.

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Thus, the final solution is .

This is a problem where elimination can be help you save a little time. You can eliminate options quickly by simplifying one power at a time and comparing your work with the answer choices.

To begin, reorder the problem so that all like terms are next to each other. When doing so, keep an eye on your signs so that you don't accidentally make a mistake.

From here, combine each pair of terms. As you do so, compare your work with the answer choices.

Eliminate any answer choices that have a different term.

Eliminate any answer choices that have a different term.

Eliminate any answer choices that have a different x term.

Eliminate any answer choices that have a different constant term.

Once you put all of your solutions together, the correct answer looks like this:

1. Factor the numerator:

2. Factor the denominator:

3. Divide the factored numerator by the factored denominator:

You can cancel out the from both the numerator and the denominator, leaving you with:

In order to divide these polynomials, you need to first factor them.

and

Now, the expression becomes

so,

The leading term of the numerator is one exponent higher than the leading term of the denominator. Thus, we know the result of the division is going to be somewhere close to . We can separate the fraction out like this:

The first term is easily seen to be , which is equal to . The second term can also be written out as:

and combining these, we get our final answer,

The degree of a polynomial is the highest exponent of the terms.

Degree 0 – constant

Degree 1 – linear

Degree 2 – quadratic

Degree 3 – cubic

Degree 4 – quartic

Multiply out the equation:

(y + 2)(y + 4)(y + 1) = z

(y2 + 2y + 4y + 8)(y + 1) = z

y3 + 2y2 + 4y2 + 8y + y2 + 2y + 4y + 8 = z

The highest exponent is y3;therefore the equation is a degree 3 cubic.

The degree of a polynomial is determined by the term with the highest degree. In this case, the first term, , has the highest degree, . The degree of a term is calculated by adding the exponents of each variable in the term.

Expand:

Step 1: Use the distributive property

Step 2: Combine like terms

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:

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:
6+12=18

When subtracting polynomials, it's helpful to remember that the "minus sign" gets distributed. It's as if the two polynomials are being added and a -1 is in front of the second polynomial.

This -1 will get multiplied to all the terms in the second polynomial that is being subtracted from the first, so it becomes . You may note that by multiplying by -1, every term in the polynomial switches its original sign. The problem then becomes:

From here, in order to simplify, because there's no equal sign, it can be deduced that we are not working toward a solution for x. The original problem is presented as an expression so an expression as an answer will be expected. In order to work towards a final simplified expression, like terms must be collected. This will provide the final answer.

Begin by distributing the subtraction of the second term in this question:

Now, you merely need to combine like terms:

To answer this question, we must distribute the to the rest of the variables , , and that are within the brackets.

To distribute a variable or number, you multiply that value with every other value within the brackets or parentheses. So, for this data:

We then simplify the expression by combining the variables we are multiplying together into expressions. For this data:

Be sure to keep all of your operations the same within the problem itself, unless the number being distributed is negative, which will then switch the signs with the brackets from positive to negative or negative to positive.

Therefore, our answer is .

To answer this question, we are solving for the values of that make this equation true.

To this, we need to get on a side by itself so we can evaluate it. To do this, we first add to both sides so that we can then begin to deal with the value. So, for this data:

can also be written as . Therefore:

Now we can divide both sides by and find the value of .

Therefore, the answer to this question is

The easisest way to solve for is to begin by plugging each pair of coordinates into the function.

Using our first point, we will plug in for and for . This gives us the equation

.

Squaring 0 gives us 0, and multiplying this by still gives 0, leaving only on the right side, such that

.

We now know the value of , and we can use this to help us find . Substituting our second set of coordinates into the function, we get

which simplifies to

.

However, since we know , we can substitute to get

subtracting 7 from both sides gives

and dividing by 4 gives our answer

.

To solve this equation, we substitute in for every instance of seen in the original equation .

Therefore the new equation would read

Now we must square the expression . To do this, you must multiply the expression by itself. Therefore:

We must now plug in our new value for into our original equation in place of .

Now we must distribute the into . To do this, you multiply each expression within the parenthesis by :

Therefore, our answer is .

Using the binomial theorem, the term containing the _x_2 _y_7 will be equal to

(2_x_)2(–y)7

=36(–4_x_2 _y_7)= -144_x_2_y_7

If the expression is expanded, then by the binomial theorem, the term is

or, equivalently, the coefficient of is

Therefore, the coefficient can be determined by setting