SAT Math - How to factor a variable

Factor to the simplest form:

Explanation

Group all the terms with the variable.

Pull out an term from parentheses.

There are no more common factors.

The correct answer is:

Factor and simplify:

$\frac{64y^{2} - 16}{8y + 4}$

$8y-4$

$8y+4$

$8y-12$

$8y$

$-4$

Explanation

$64y^{2} - 16$ is a difference of squares.

The difference of squares formula is $a^{2} - b^{2} = (a - b)(a + b)$ .

Therefore, $\frac{64y^{2} - 16}{8y + 4}$ = $\frac{(8y + 4)(8y - 4)}{8y + 4} = 8y - 4$ .

A group of scientists form a global collective of temperature and climate data. U.S. temperature measurements are in Fahrenheit and must be converted to Celsius. If the average spring temperature for New York was $50^{\circ}$ Fahrenheit, what is the temperature value in Celsius?

The Fahrenheit to Celsius conversion equation is as follows:

$T_{Celsius}=(T_{Fahrenheit}-32)\cdot \frac{5}{9}$

$10^{\circ}:C$

$12^{\circ}:C$

$15^{\circ}:C$

$8^{\circ}:C$

$6^{\circ}:C$

Explanation

You can solve this problem by substituting in $50^{\circ}$ for $T_{Fahrenheit}$ and solving for $T_{Celsius}$ :

$T_{Celsius}=(50-32)\cdot \frac{5}{9}$

$T_{Celsius}=18\cdot \frac{5}{9}$

$T_{Celsius}=\frac{18}{1}\cdot\frac{5}{9}=\frac{18\cdot5}{1\cdot9}=\frac{90}{9}=10$

That means that $50^{\circ}$ Fahrenheit is the same as $10^{\circ}$ Celsius.

Factor 9_x_2 + 12_x_ + 4.

(3_x_ + 2)(3_x_ + 2)

(3_x_ + 2)(3_x_ – 2)

(3_x_ – 2)(3_x_ – 2)

(9_x_ + 4)(9_x_ – 4)

(9_x_ + 4)(9_x_ + 4)

Explanation

Nothing common cancels at the beginning. To factor this, we need to find two numbers that multiply to 9 * 4 = 36 and sum to 12. 6 and 6 work.

So 9_x_2 + 12_x_ + 4 = 9_x_2 + 6_x_ + 6_x_ + 4

Let's look at the first two terms and last two terms separately to begin with. 9_x_2 + 6_x_ can be simplified to 3_x_(3_x_ + 2) and 6_x_ + 4 can be simplified into 2(3_x_ + 2). Putting these together gets us

9_x_2 + 12_x_ + 4

= 9_x_2 + 6_x_ + 6_x_ + 4

= 3_x_(3_x_ + 2) + 2(3_x_ + 2)

= (3_x_ + 2)(3_x_ + 2)

This is as far as we can factor.

When is factored, it can be written in the form , where , , , , , and are all integer constants, and .

What is the value of ?

Explanation

Let's try to factor x2 – y2 – z2 + 2yz.

Notice that the last three terms are very close to y2 + z2 – 2yz, which, if we rearranged them, would become y2 – 2yz+ z2. We could factor y2 – 2yz+ z2 as (y – z)2, using the general rule that p2 – 2pq + q2 = (p – q)2 .

So we want to rearrange the last three terms. Let's group them together first.

x2 + (–y2 – z2 + 2yz)

If we were to factor out a –1 from the last three terms, we would have the following:

x2 – (y2 + z2 – 2yz)

Now we can replace y2 + z2 – 2yz with (y – z)2.

x2 – (y – z)2

This expression is actually a differences of squares. In general, we can factor p2 – q2 as (p – q)(p + q). In this case, we can substitute x for p and (y – z) for q.

x2 – (y – z)2 = (x – (y – z))(x + (y – z))

Now, let's distribute the negative one in the trinomial x – (y – z)

(x – (y – z))(x + (y – z))

(x – y + z)(x + y – z)

The problem said that factoring x2 – y2 – z2 + 2yz would result in two polynomials in the form (ax + by + cz)(dx + ey + fz), where a, b, c, d, e, and f were all integers, and a > 0.

(x – y + z)(x + y – z) fits this form. This means that a = 1, b = –1, c = 1, d = 1, e = 1, and f = –1. The sum of all of these is 2.

The answer is 2.

If $\dpi{100} \small \frac{x^{2}-9}{x+3}=5$ , and $\dpi{100} \small x eq -3$ , what is the value of $\dpi{100} \small x$ ?

–8

–6

Explanation

The numerator on the left can be factored so the expression becomes $\dpi{100} \small \frac{\left ( x+3 \right )\times \left ( x-3 \right )}{\left ( x+3 \right )}=5$ , which can be simplified to $\dpi{100} \small \left ( x-3 \right )=5$

Then you can solve for $\dpi{100} \small x$ by adding 3 to both sides of the equation, so $\dpi{100} \small x=8$

Factor:

$-12x^2+27$

$-3(2x+3)(2x-3)$

$-3(4x^{2}-9)$

$(2x+3)(2x-3)$

$(2x+3)(2x+3)$

$-3(2x+3)(2x+3)$

Explanation

We can first factor out $-3$ :

$-3(4x^{2}-9)$

This factors further because there is a difference of squares:

$-3(2x+3)(2x-3)$

Solve for x:

$\small x^2+3x+2=0$

$\dpi{100} \small x=-2\ or-1$

$\dpi{100} \small x=2\ or\ 1$

$\dpi{100} \small x=-2\ or\ 1$

$\dpi{100} \small x=2\ or-1$

Explanation

First, factor.

$\small x^2+3x+2=(x+2)(x+1)=0$

Set each factor equal to 0

$\small x+2=0; x=-2$

$\small x+1=0; x=-1$

Therefore,

$\dpi{100} \small x=-2\ or-1$

Factor the following variable

(x2 + 18x + 72)

(x + 18) (x + 72)

(x + 6) (x + 12)

(x – 6) (x – 12)

(x + 6) (x – 12)

(x – 6) (x + 12)

Explanation

You need to find two numbers that multiply to give 72 and add up to give 18

easiest way: write the multiples of 72:

1, 72

2, 36

3, 24

4, 18

6, 12: these add up to 18

(x + 6)(x + 12)

A semi truck unloaded weighs . When the trailer compartment is loaded to half capacity with soda, the semi weighs . What will the truck weigh when the compartment is loaded to $\frac{3}{4}$ capacity with the same kind of soda?