Variables - PSAT Math

Card 0 of 532

Question

Factor the following variable

(x2 + 18x + 72)

Answer

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)

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Question

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

What is the value of ?

Answer

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.

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Question

Factor 9_x_2 + 12_x_ + 4.

Answer

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.

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Question

Factor and simplify:

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

Answer

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

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

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

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Question

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

Answer

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 $\dpi{100} \small x=8$

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Question

Factor:

$-12x^2+27$

Answer

We can first factor out $-3$ :

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

This factors further because there is a difference of squares:

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

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Question

Solve for x:

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

Answer

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,

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

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Question

Answer

Use the distributive property:

Combine like terms:

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Question

Find the product:

Answer

Find the product:

Step 1: Use the Distributive Property

Step 2: Combine like terms

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Question

Find the product:

Answer

Find the product:

Use the distributive property:

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Question

Answer

Use the distributive property:

Combine like terms:

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Question

Find the product:

Answer

Find the product:

Step 1: Use the Distributive Property

Step 2: Combine like terms

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Question

Find the product:

Answer

Find the product:

Use the distributive property:

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Question

Find the degree of the polynomial

$\small - x^{2}+3x - 2x^{5} - x^{3} +4$

Answer

The degree of the polynomial is the largest degree of any one of it's individual terms.

$\small - x^{2}+3x - 2x^{5} - x^{3} +4$

The degree of $\small -x^2$ is $\small 2$

The degree of $\small 3x$ is $\small 1$

The degree of $\small -2x^5$ is $\small 5$

The degree of $\small -x^3$ is $\small 3$

The degree of $\small 4$ is $\small 0$

$\small 5$ is the largest degree of any one of the terms of the polynomial, and so the degree of the polynomial is $\small 5$ .

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Question

Give the degree of the polynomial

$777a^{12}b^{14}c^{16}d^{18}$

Answer

The polynomial has one term, so its degree is the sum of the exponents of the variables:

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Question

Give the degree of the polynomial

$x^{50}y^{30} - x^{70}y^{25}+ x^{45}y ^{45} - x^{30}y ^{70}$

Answer

The degree of a polynomial in more than one variable is the greatest degree of any of the terms; the degree of a term is the sum of the exponents. The degrees of the terms in the given polynomial are:

The degree of the polynomial is the greatest of these degrees, 100.

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Question

Give the degree of the polynomial

$y^{44} + y^{20} - y^{10} + y^{100}$

Answer

The degree of a polynomial in one variable is the greatest exponent of any of the powers of the variable. The terms have as their exponents, in order, 44, 20, 10, and 100; the greatest of these is 100, which is the degree.

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Question

Give the degree of the polynomial

$400x^{10}- 300x^{20} + 200x^{30}-100 x^{40}$

Answer

The degree of a polynomial in one variable is the greatest exponent of any of the powers of the variable. The terms have as their exponents, in order, 10, 20, 30, and 40; 40 is the greatest of them and is the degree of the polynomial.

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Question

Which of these polynomials has the greatest degree?

Answer

The degree of a polynomial is the highest degree of any term; the degree of a term is the exponent of its variable or the sum of the exponents of its variables, with unwritten exponents being equal to 1. For each term in a polynomial, write the exponent or add the exponents; the greatest number is its degree. We do this with all four choices:

$\underline{6x^{3}y^{2} + 4 xy^{3} - 100}$ :

$6x^{3}y^{2}: 3 + 2 = 5$

$4 xy^{3}: 1 + 3 = 4$

A constant term has degree 0.

The degree of this polynomial is 5.

$\underline{7xy^{4} - 4x^{2}+ y^{3}}$

$7xy^{4} : 1 + 4 = 5$

$4x^{2} : 2$

$y^{3}: 3$

The degree of this polynomial is 5.

$\underline{8x ^{4}y - 4 y^{2} + 9x^{3}y}$

$8x ^{4}y: 4 + 1= 5$

$4 y^{2} : 2$

$9x^{3}y: 3 + 1= 4$

The degree of this polynomial is 5.

$\underline{-5 x^{2}y^{2}+ 7x^{3}y^{2}+ 8x}$

$-5 x^{2}y^{2} : 2 + 2 = 4$

$7x^{3}y^{2} : 3 + 2 = 5$

The degree of this polynomial is 5.

All four polynomials have the same degree.

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Question

Which of the following monomials has degree 999?

Answer

The degree of a monomial term is the sum of the exponents of its variables, with the default being 1.

For each monomial, this sum - and the degree - is as follows:

$444x^{999}yz$ :

$333x^{999}y^{999}z^{999}$ :

$999x^{4}y^{5}z^{6}$ : (note - 999 is the coefficient)

$100x^{333}y^{333}z^{333}$ :

$100x^{333}y^{333}z^{333}$ is the correct choice.

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