Algebra - Variables | Practice Hub

Explanation

To solve this problem, simply add the terms with like exponents and variables:

$\(2x^2+7x+4) + (x^2-3x+2) \= (2x^2 +x^2) + (7x-3x) + (4+2) \= 3x^2 - 4x +6$

Thus, is our answer.

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 $\left ( 0.2x+ 5 \right )^{7}$ .

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

$C(7,5) \cdot 0.2^{5} \cdot 5 ^{7 - 5}$

$= C(7,5) \cdot 0.2 ^{5} \cdot 5 ^{2}$

$= 21\cdot 0.00032 \cdot 25$

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.

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

Explanation

Let's split the four terms into two groups, and find the GCF of each group.

First group:

Second group:

The GCF of the first group is . When we divide the first group's terms by , we get: .

The GCF of the second group is . When we divide the second group's terms by , we get: .

We can rewrite the original expression,

as,

The common factor for BOTH of these terms is .

Dividing both sides by gives us:

Give the coefficient of $x^{2}$ in the product

$\left ( x+ 0.4\right ) (x - 0.2) (3x-0.7)$ .

Explanation

While this problem can be answered by multiplying the three binomials, it is not necessary. There are three ways to multiply one term from each binomial such that two terms and one constant are multiplied; find the three products and add them, as follows:

$\left (\underline{ x}+ 0.4\right ) (\underline{x} - 0.2) (3x\underline{-0.7})$

$x \cdot x \cdot (-0.7) = -0.7x^{2}$

$\left (\underline{ x}+ 0.4\right ) (x \underline{-0.2})(\underline{3x}-0.7)$

$x \cdot (-0.2) \cdot 3x= -0.6x^{2}$

$\left (x+ \underline{0.4}\right ) (\underline{x} - 0.2)(\underline{3x}-0.7)$

$0.4 \cdot x \cdot 3x = 1.2 x^{2}$

Add: $-0.7x^{2}+ (-0.6x^{2})+ 1.2x^{2} = -0.1x^{2}$ .

The correct response is .

Give the coefficient of $x^{4}$ in the binomial expansion of $\left (6x+ \frac{1}{6}\right )^{7}$ .

$\frac{35}{6}$

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 $x^{4}$ coefficient can be determined by setting

$A = 6, B = \frac{1}{6}, n = 7, k= 4$ :

$C(7,4) \cdot 6 ^{4} \cdot\left ( \frac{1}{6} \right ) ^{7-4}$

$=C(7,4) \cdot 6 ^{4} \cdot\left ( \frac{1}{6} \right ) ^{3}$

$=35 \cdot 1,296 \cdot \frac{1}{216}$

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 five is the highest exponent above, it is also the degree of the polynomial.

Expand:

$3x^{2}-25x-18$

$3x^{2}-29x-18$

$3x^{2}-18x-29$

$3x^{2}-18x-25$

None of the other answers

Explanation

To multiple these binomials, you can use the FOIL method to multiply each of the expressions individually.This will give you

or $3x^{2}-25x-18$ .

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 nine is the highest exponent above, it is also the degree of the polynomial.

Variables

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