Variables

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Algebra › Variables

Questions 1 - 10

Factor the following expression:

not factorable

Explanation

We are going to use factoring by grouping to factor this expression.

The expression:

has the form:

In factoring by grouping, we want to split that B value into two smaller values a and b so we end up with this expression:

The rules are:

and

For our problem A=6 B=-13 C=6

So we have:

Another way of saying this is that we are looking for factors of 36 that add up to -13

The values that work are:

Plugging those values in:

Now let's group and factor out from both groups:

Finally let's factor out a (2x-3) from both terms: and we get our answer:

Multiply the expressions:

$\small (x^{2} - 2x + 7) (x^{2} - 2x - 7)$

$\small x^{4}-4x^{3}+4x^2-49$

$\small \small x^{4}-4x^{3}+4x^2+49$

$\small \small \small x^{4}-4x^{3}-4x^2-49$

$\small \small \small \small \small x^{4}+4x^{3}-4x^2+49$

$\small \small \small \small x^{4}+4x^{3}-4x^2-49$

Explanation

You can look at this as the sum of two expressions multiplied by the difference of the same two expressions. Use the pattern

$\small (A+B)(A-B) = (A^{2}-B^{2})$ ,

where $\small A = x^{2} - 2x$ and $\small B = 7$ .

$\small \small \small (x^{2} - 2x + 7) (x^{2} - 2x - 7) = (x^{2} - 2x )^{2} - 7^{2} = (x^{2} - 2x )^{2} - 49$

To find $\small (x^{2} - 2x )^{2}$ , you use the formula for perfect squares:

$\small (A-B)^{2} = (A^{2}-2AB+B^{2})$ ,

where $\small \small A = x^{2}$ and $\small B = 2x$ .

$\small \small \small (x^{2}-2x)^{2} = ((x^{2})^{2}-2x^{2}(2x)+(2x)^{2}) = x^{4}-4x^{3}+4x^2$

Substituting above, the final answer is $\small x^{4}-4x^{3}+4x^2-49$ .

Find the degree of the following polynomial:

Explanation

Find the degree of the following polynomial:

To find the degree of a polynomial, we simply need to look at its highest exponent.

The degree will be equal to the highest exponent, which in this case, is 7

$g(x)=6x^{\color{DarkGreen} 7}-5x^5+144x^3-1200x+33$

So, the correct answer is 7

What is the degree of the following polynomial?

Explanation

To find the degree of a polynomial, simply find the highest exponent in the expression. As seven is the highest exponent above, it is also the degree of the polynomial.

Multiply:

$\left (x^{2} + 2x + 7 \right ) \left (x^{2} + x - 6 \right )$

$x^{4} + 3 x^{3} +3x^{2} - 5x - 42$

$x^{4} + 3 x^{3} -x^{2} - 5x - 42$

$x^{4} + 3 x^{3} -15x^{2} - 19x - 42$

$x^{4} + 3 x^{3} -x^{2} - 19x - 42$

$x^{4} + 3 x^{3} + 19x^{2} - 5x - 42$

Explanation

$\left (x^{2} + 2x + 7 \right ) \left (x^{2} + x - 6 \right )$

$= x^{2}\left (x^{2} + x - 6 \right ) + 2x \left (x^{2} + x - 6 \right ) + 7 \left (x^{2} + x - 6 \right )$

$= x^{2} \cdot x^{2} + x^{2} \cdot x - x^{2} \cdot 6 + 2x \cdot x^{2} + 2x \cdot x - 2x \cdot 6 + 7 \cdot x^{2} + 7 \cdot x - 7 \cdot 6$

$= x^{4} + x^{3} - 6x^{2} + 2x^{3} + 2x^{2} - 12x + 7x^{2} +7x - 42$

$= x^{4} + x^{3} + 2x^{3} - 6x^{2} + 2x^{2} + 7x^{2} - 12x +7x - 42$

$= x^{4} + 3 x^{3} +3x^{2} - 5x - 42$

Give the coefficient of $x^{5}$ in the binomial expansion of $\left ( 2x+ 0.5\right )^{8}$ .

Explanation

If the expression $\left ( A x + B\right )^{n}$ is expanded, then by the binomial theorem, the $x^{k}$ term is

$C(n, k) \cdot\left ( Ax \right )^{k} \cdot B ^{n - k}$

$= C(n, k) \cdot A ^{k} \cdot B ^{n - k} \cdot x^{k}$

or, equivalently, the coefficient of $x^{k}$ is

$C(n, k) \cdot A ^{k} \cdot B ^{n - k}$

Therefore, the $x^{5}$ coefficient can be determined by setting

$C(8,5) \cdot 2^{5} \cdot 0.5 ^{8-5}$

$=C(8,5) \cdot 2^{5} \cdot 0.5 ^{3}$

$= 56\cdot 32 \cdot 0.125$

Factor the following polynomial expression completely, using the "factor-by-grouping" method.

Explanation

Separate the four terms into two groups, and then find the GCF of each group.

First group:

Second group:

The GCF of the first group is . Factoring out from the terms in the first group gives us:

The GCF of the second group is . Factoring out from the terms in the second group gives us:

We can rewrite the original expression,

as,

We can factor this as:

Multiply:

Explanation

In order to solve, we will need to multiply each term of the first trinomial with all the terms of the second trinomial. Sum all the terms together.

Add all of the terms and combine like terms.

The answer is:

Multiply the expressions:

$\small (x^{2} - 2x + 7) (x^{2} - 2x - 7)$

$\small x^{4}-4x^{3}+4x^2-49$

$\small \small x^{4}-4x^{3}+4x^2+49$

$\small \small \small x^{4}-4x^{3}-4x^2-49$

$\small \small \small \small \small x^{4}+4x^{3}-4x^2+49$

$\small \small \small \small x^{4}+4x^{3}-4x^2-49$

Explanation

You can look at this as the sum of two expressions multiplied by the difference of the same two expressions. Use the pattern

$\small (A+B)(A-B) = (A^{2}-B^{2})$ ,

where $\small A = x^{2} - 2x$ and $\small B = 7$ .

$\small \small \small (x^{2} - 2x + 7) (x^{2} - 2x - 7) = (x^{2} - 2x )^{2} - 7^{2} = (x^{2} - 2x )^{2} - 49$

To find $\small (x^{2} - 2x )^{2}$ , you use the formula for perfect squares:

$\small (A-B)^{2} = (A^{2}-2AB+B^{2})$ ,

where $\small \small A = x^{2}$ and $\small B = 2x$ .

$\small \small \small (x^{2}-2x)^{2} = ((x^{2})^{2}-2x^{2}(2x)+(2x)^{2}) = x^{4}-4x^{3}+4x^2$

Substituting above, the final answer is $\small x^{4}-4x^{3}+4x^2-49$ .

Solve for in terms of .

Explanation

First, add 8y to both sides of the equation, cancelling out -8y and isolating x to a value in terms of y:

4x - 8y = 32

+8y +8y

4x = 32 + 8y

Then divide both sides of the equation by 4, providing the x-value in terms of y:

x = 8 + 2y

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