PSAT Math : How to factor an equation

Study concepts, example questions & explanations for PSAT Math

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Example Questions

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Example Question #1 : Equations / Inequalities

Factor 4x3 – 16x

Possible Answers:

4x(x + 2)(x + 2)

4x(x + 2)(x – 2)

cannot be factored

4x(x2 – 4)

4x(x – 2)(x – 2)

Correct answer:

4x(x + 2)(x – 2)

Explanation:

First pull out any common terms: 4x3 – 16x = 4x(x2 – 4)

x2 – 4 is a difference of squares, so we can also factor that further. The difference of squares formula is a2 – b2 = (a – b)(a + b). Here a = x and b = 2. So x2 – 4 = (x – 2)(x + 2).

Putting everything together, 4x3 – 16x = 4x(+ 2)(– 2).

Example Question #2 : Equations / Inequalities

Factor the following equation. 

x– 16

Possible Answers:

(x + 4)(x – 4)

(x – 4)(x – 4)

(x2)(4 – 2) 

(x)(x – 4)

(x + 4)(x + 4)

Correct answer:

(x + 4)(x – 4)

Explanation:

The correct answer is (x + 4)(x – 4) 

We neen to factor x– 16 to solve. We know that each parenthesis will contain an x to make the x2. We know that the root of 16 is 4 and since it is negative and no value of x is present we can tell that one 4 must be positive and the other negative. If we work it from the multiple choice answers we will see that when multiplying it out we get x+ 4x – 4x – 16. 4x – 4x cancels out and we are left with our answer. 

Example Question #1 : How To Factor An Equation

If x3 – y3 = 30, and x2 + xy + y2 = 6, then what is x2 – 2xy + y2?

Possible Answers:

5

24

cannot be determined

25

180

Correct answer:

25

Explanation:

First, let's factor x3 – y3 using the formula for difference of cubes.

x3 – y= (x – y)(x2 + xy + y2)

We are told that x2 + xy + y2 = 6. Thus, we can substitute 6 into the above equation and solve for x – y.

(x - y)(6) = 30.

Divide both sides by 6.

x – y = 5.

The original questions asks us to find x2 – 2xy + y2. Notice that if we factor x2 – 2xy + y2 using the formula for perfect squares, we obtain the following:

x2 – 2xy + y= (x – y)2.

Since we know that (x – y) = 5, (x – y)2 must equal 52, or 25.

Thus, x2 – 2xy + y= 25.

The answer is 25.

Example Question #1 : How To Factor An Equation

if x – y = 4 and x2 – y = 34, what is x?

Possible Answers:

10

15

6

12

9

Correct answer:

6

Explanation:

This can be solved by substitution and factoring.

x2 – y = 34 can be written as y = x2 – 34 and substituted into the other equation: x – y = 4 which leads to x – x2 + 34 = 4 which can be written as x2 – x – 30 = 0.

x2 – x – 30 = 0 can be factored to (x – 6)(x + 5) = 0 so x = 6 and –5 and because only 6 is a possible answer, it is the correct choice.

Example Question #2 : How To Factor An Equation

If x2 + 2ax + 81 = 0.  When a = 9, what is the value of x?

Possible Answers:

3

0

9

–18

–9

Correct answer:

–9

Explanation:

When a = 9, then x2 + 2ax + 81 = 0 becomes 

x2 + 18x + 81 = 0.  

This equation can be factored as (x + 9)2 = 0.

Therefore when a = 9, x = –9.

Example Question #4 : How To Factor An Equation

If f(x) has roots at x = –1, 0 and 2, which of the following could be the equation for f(x)?

Possible Answers:

f(x) = x– x– 2x

f(x) = x2 + x – 2

f(x) = x– x – 2

f(x) = x4 + x3 – 2x2

f(x) = x– x+ 2x

Correct answer:

f(x) = x– x– 2x

Explanation:

In general, if a function has a root at x = r, then (x – r) must be a factor of f(x). In this problem, we are told that f(x) has roots at –1, 0 and 2. This means that the following are all factors of f(x):

(x – (–1)) = x + 1

(x – 0) = x

and (x – 2).

This means that we must look for an equation for f(x) that has the factors (+ 1), x, and (x – 2).

We can immediately eliminate the function f(x) = x2 + x – 2, because we cannot factor an x out of this polynomial. For the same reason, we can eliminate f(x) = x– x – 2.

Let's look at the function f(x) = x– x+ 2x. When we factor this, we are left with x(x2 – x + 2). We cannot factor this polynomial any further. Thus, x + 1 and x – 2 are not factors of this function, so it can't be the answer.

Next, let's examine f(x) = x4 + x3 – 2x.

We can factor out x2.

x(x+ x – 2)

When we factor x+ x – 2, we will get (x + 2)(x – 1). These factors are not the same as x – 2 and x + 1. 

The only function with the right factors is f(x) = x– x– 2x.

When we factor out an x, we get (x– x – 2), which then factors into (x – 2)(+ 1). Thus, this function has all of the factors we need.

The answer is f(x) = x– x– 2x.

Example Question #1 : How To Factor An Equation

Factor 36x2 – 49y2.

Possible Answers:

(6x + 7y)(6x + 7y)

(6x – 7y)(6x – 7y)

cannot be factored

6x2 – 7y2

(6x + 7y)(6x – 7y)

Correct answer:

(6x + 7y)(6x – 7y)

Explanation:

This is a difference of squares. The difference of squares formula is a2 – b2 = (a + b)(a – b). In this problem, a = 6x and b = 7y.

So 36x2 – 49y= (6x + 7y)(6x – 7y).

Example Question #4 : How To Factor An Equation

Solve for x:

\dpi{100} \small x^{2}-2x-48 = 0

Possible Answers:

\dpi{100} \small x=8,-6

Correct answer:

\dpi{100} \small x=8,-6

Explanation:

Find two numbers that add to \dpi{100} \small -2 and multiply to \dpi{100} \small -48

Factors of \dpi{100} \small 48

\dpi{100} \small 1,2,3,4,6,8,12,16,24,48

You can use \dpi{100} \small -8 +6 =-2

\dpi{100} \small (x-8)(x+6) = 0

Then make each factor equal 0.

\dpi{100} \small x-8 = 0 and \dpi{100} \small x+6 = 0

\dpi{100} \small x= 8 and \dpi{100} \small x=-6

Example Question #9 : Equations / Inequalities

Find the roots of f(x)=x^2+2x-3

Possible Answers:

x=1,3

x=0,3

x=-1,3

x=1,-3

x=2,1

Correct answer:

x=1,-3

Explanation:

Factoring yields (x+3)(x-1) giving roots of -3 and 1.

Example Question #5 : How To Factor An Equation

\frac{-3-2x+x^{2}}{x-3}

 

Find the root of the equation above.

Possible Answers:

x=-1

x=0

x=1

x=3

x=2

Correct answer:

x=-1

Explanation:

The numerator can be factored into (x-3)(x+1).

Therefore, it can cancel with the denominator. So x+1=0 imples x=-1.

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