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ACT Math : Polynomial Operations

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Example Questions

Example Question #611 : Algebra

Simplify:

3x^2 + 5x +10 - (10x^2-15x-3)

Possible Answers:

13x^2 +13

-7x^2 + 20x +13

-17x^2+7

3x^2 +5x +7

-7x^2 - 10x +7

Correct answer:

-7x^2 + 20x +13

Explanation:

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

3x^2 + 5x +10 -10x^2+15x+3

Now, you merely need to combine like terms:

3x^2 -10x^2 + 5x +15x +10 +3

-7x^2 + 20x +13

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Example Question #12 : Polynomials

If $f(x) = 2x^{2} - 3$ , then what does f(x + a) equal?

Possible Answers:

$2x^{2} - 4ax + 2a^{2} - 3$

$4x^{2} + 8ax + 4a^{2} - 3$

$2x^{2} + 4ax + 2a^{2} - 3$

$4x^{2} + 4ax + 4a^{2} - 3$

$2x^{2} + 2ax + 2a^{2} - 3$

Correct answer:

$2x^{2} + 4ax + 2a^{2} - 3$

Explanation:

To solve this equation, we substitute (x+a) in for every instance of seen in the original equation $f(x)=2x^{2}-3$ .

Therefore the new equation would read

$f(x)=2x^{2}-3\rightarrow f(x+a) = 2(x + a)^{2} -3$

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

$(x+a)^{2}=(x+a)\cdot(x+a)$

$(x+a)\cdot(x+a)=(x\cdot x)+(x\cdot a)+(x\cdot a)+(a\cdot a)$

$(x\cdot x)+(x\cdot a)+(x\cdot a)+(a\cdot a)=x^{2}+2ax+a^{2}$

We must now plug in our new value for $(x+a)^{2}$ into our original equation in place of $x^{2}$ .

$f(x)=2(x^{2}+2ax+a^{2})-3$

Now we must distribute the into $(x^{2}+2ax+a^{2})$ . To do this, you multiply each expression within the parenthesis by :

$2(x^{2}+2ax+a^{2})=2x^{2}+4ax+2a^{2}$

Therefore, our answer is $f(x)=2x^{2}+4ax+2a^{2}-3$ .

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Example Question #11 : Polynomials

The expression a[(b-c) +d] is equivalent to which of the following?

Possible Answers:

b-c+ad

ab-ac-ad

ab-ac+ad

ab+ac+ad

ab+ac-ad

Correct answer:

ab-ac+ad

Explanation:

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:

$a[(b-c)+d] = [[(a\cdot b)-(a\cdot c)] + (a\cdot d)]$

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

$[[(a\cdot b)-(a\cdot c)] + (a\cdot d)] = ab-ac+ad$

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 ab-ac+ad .

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Example Question #14 : Polynomials

Solve the equation $x^{2} - 25x = 0$

Possible Answers:

Correct answer:

Explanation:

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 25x to both sides so that we can then begin to deal with the $x^{2}$ value. So, for this data:

$x^{2}-25x=0\rightarrow x^{2}=25x$

$x^{2}$ can also be written as $x\cdot x$ . Therefore:

$x^{2}=25x\rightarrow x\cdot x=25x$

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

$\frac{x\cdot x}{x}=\frac{25x}{x}\rightarrow x=25$

Therefore, the answer to this question is x=25

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Example Question #11 : Polynomial Operations

Simplify the following expression.

(2x^3+6x^2-3x+1) + (x^3-2x^2+4x-3)

Possible Answers:

3x^3+2x^2+2x+2

3x^3+3x^2+x+4

3x^4-4x^2+7x-2

3x^3+4x^2+x-2

2x^3+6x^2+2x+2

Correct answer:

3x^3+4x^2+x-2

Explanation:

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

2x^3+6x^2-3x+1

x^3-2x^2+4x-3

____________________

2x^3+x^3 = 3x^3

6x^2-2x^2 =4x^2

-3x+4x=x

1-3=-2

Thus, the final solution is 3x^3+4x^2+x-2 .

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Example Question #12 : Polynomials

What is the value of when 5x-5y=5y-5x

Possible Answers:

x-y=0

x=y

x+y=0

25x=y

25y=x

Correct answer:

x=y

Explanation:

5x-5y=5y-5x

In adding to both sides:

5x-5y+5x=5y-5x+5x

10x-5y=5y

. . .and adding to both sides:

10x-5y+5y=5y+5y

. . .the variables are isolated to become:

10x=10y

After dividing both sides by , the equation becomes:

x=y

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Example Question #2 : How To Add Polynomials

Add the following polynomials:

$(5x^{3}+31x^{2}-17x-6)+(-2x^{3}+9x^{2}+34x-12)$

Possible Answers:

$3x^{3} + 40x^{2} +17x -18$

$-3x^{3} + 40x^{2} -17x +6$

$3x^{3} + 22x^{2} +10x -18$

$-7x^{3} + 40x^{2} +17x -18$

$7x^{3} + 40x^{2} +17x +6$

Correct answer:

$3x^{3} + 40x^{2} +17x -18$

Explanation:

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.

$(5x^{3}-2x^{3}) + (31x^{2}+9x^{2}) +(-17x+34x) +(-6-12)$

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

$(5x^{3}-2x^{3})=3x^{3} }$ Eliminate any answer choices that have a different x^3 term.

$(31x^{2}+9x^{2})=40x^2$ Eliminate any answer choices that have a different x^2 term.

(-17x+34x)=17x Eliminate any answer choices that have a different x term.

(-6-12)= -6+-12=-18 Eliminate any answer choices that have a different constant term.

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

$3x^{3} + 40x^{2} +17x -18$

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ACT Math : Polynomial Operations

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #611 : Algebra

Example Question #12 : Polynomials

Example Question #11 : Polynomials

Example Question #14 : Polynomials

Example Question #11 : Polynomial Operations

Example Question #12 : Polynomials

Example Question #2 : How To Add Polynomials

All ACT Math Resources

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