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ACT Math : Variables

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Example Questions

Example Question #101 : Variables

Which expression is equivalent to the following polynomial:

x^2-81

Possible Answers:

(x+80)(x+1)

(x-9)^2

(x-90)(x+9)

(x-9)(x+9)

x(x-81)

Correct answer:

(x-9)(x+9)

Explanation:

This question calls for us to factor the polynomial into two binomials. Since the first term is x^2 and the last term is a number without a variable, we know that how answer will be of the form (x+ a)(x+ b) where a and b are positive or negative numbers.

To find a and b we look at the second and third term. Since the second term is not present, we know a+b=0 . (The x comes from a and b multiplying by x and then adding with each other).

The term tells us that a*b=-81 . Using these two pieces of information we can look at possible values. The third term tells us that -81 & 1, 81 & -1, and -9 & 9 are the possible pairs. Now we can look and see which one adds up to make 0.

This gives us the pair -9 & 9 and we plug that into the equation as a and b to get our final answer.

(x-9)(x+9)

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Example Question #101 : Variables

Factor the following polynomial:

x^2 -14x + 33

Possible Answers:

(x-33)(x+1)

(x-11)(x-3)

(x+11)(x-3)

(x-11)(x+3)

(x-7)(x+2)

Correct answer:

(x-11)(x-3)

Explanation:

To factor a polynomial of the form ax^2 + bx + c begin by factoring both and .

Since a=1 we are done.

When you factor your two factors need to add together to get .

Since:

-11*-3 = 33 and -11 +-3=-14 the two factors we want are and .

Simply plug them into the parentheses and you have:

(x-11)(x-3)

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Example Question #1 : How To Factor A Variable

Two consecutive positive multiples of five have a product of 300. What is their sum?

Possible Answers:

Correct answer:

Explanation:

Define the variables as x = 1st number and x + 5 = 2nd number, so the product is given as x(x + 5) = 300, which becomes x² + 5x – 300 = 0.

Factoring results in (x + 20)(x – 15) = 0, so the positive answer is 15, making the second number 20.

The sum of the two numbers is 35.

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Example Question #1 : Factoring

Factor 12x³y⁴+ 156x²y³

Possible Answers:

12x²y³

x²y³(xy + 13)

12xy(xy + 13)

12x²y³(xy + 13)

Correct answer:

12x²y³(xy + 13)

Explanation:

The common factors are 12, x², and y³.

So 12x²y³(xy + 13)

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Example Question #2 : Factoring

Solve for all solutions of $\dpi{100} \small x$ :

$\dpi{100} \small 2x^{2}-10x=x^{2}-24$

Possible Answers:

$\dpi{100} \small 3,8$

$\dpi{100} \small -4,6$

$\dpi{100} \small -4,-6$

$\dpi{100} \small 3,-8$

$\dpi{100} \small 4,6$

Correct answer:

$\dpi{100} \small 4,6$

Explanation:

First move all of the variables to the left side of the equation. Combine similar terms, and set the equation equal to zero. Then factor the equation to get $\dpi{100} \small (x-4)(x-6)=0$

Thus the solutions of $\dpi{100} \small x$ are 4 and 6.

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Example Question #2 : How To Factor A Variable

Simplify:

$\frac{x^2+6x+9}{x+3}$

Possible Answers:

x+3

x^2+6x+9

x-3

x^2+5x+6

Correct answer:

x+3

Explanation:

$x^{2}+6x+9$ factors to (x+3)(x+3)

One (x+3) cancels from the bottom, leaving (x+3)

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Example Question #2 : How To Factor A Variable

Factor: $x^{2}+2x-24$

Possible Answers:

(x+1)(x-3)

(x+6)(x-4)

(2x+4)(x-6)

(x+4)(x-6)

(x+6)(x+4)

Correct answer:

(x+6)(x-4)

Explanation:

$x^{2}+2x-24$

In the form of $ax^{2}+bx+c$ you must find two numbers which add to give you and multiply to give you and then put them in the form of ( + number) ( + number)

$6\ast-4=24$

6+-4=2

Therefore (x+6)(x-4) is the answer.

To check, multiply the two expressions out and it should equal $x^{2}+2x-24$

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Example Question #3 : Factoring

Factor the following expression:

3x^2 +12x

Possible Answers:

The expression is already simplified as much as possible.

3x^2(1+4)

3x(x+4)

3(x^2+4x)

x(3x+12)

Correct answer:

3x(x+4)

Explanation:

To factor an expression we look for the greatest common factor.

Remember that

$x^2 = x\cdot x$

Thus:

$\\ 3x^2+12x \\ \\= 3\cdot x\cdot x+2\cdot 2\cdot 3\cdot x\\ \\=3x(x+2\cdot 2)\\ \\=3x(x+4)$

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Example Question #1 : Factoring

Factor the following expression:

x^2-13x+40

Possible Answers:

(x+8)(x+5)

(x+10)(x+4)

(x-8)(x-5)

(x-10)(x-4)

Correct answer:

(x-8)(x-5)

Explanation:

To factor, you are looking for two factors of 40 that add to equal 13.

Factors of 40 include: (1, 40), (2, 20), (4, 10), (5, 8). Of these factors which two will add up to 13?

Also, since the first sign (-) and the second sign is (+) this tells us both binomials will be negative. This is because two negatives multiplied together will result in the positive third term, while two negatives added together will result in a larger negative number.

Thus,

(x-8)(x-5)

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ACT Math : Variables

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #101 : Variables

Example Question #101 : Variables

Example Question #1 : How To Factor A Variable

Example Question #1 : Factoring

Example Question #2 : Factoring

Example Question #2 : How To Factor A Variable

Example Question #2 : How To Factor A Variable

Example Question #3 : Factoring

Example Question #1 : Factoring

All ACT Math Resources

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