Question

Perform the indicated operations and simplify

(a)$3(x+6)+4(2x-5)$

(b)$(x+3)(4x-5)$

(c)$(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})$

(d)${(2x+3)}^2$

(e)${(x+2)}^3$

Short answer

(a)$11x-2$

(b)$4x^2+7x-15$

(c)$a-b$

(d)$4x^2+12x+9$

(e)${(x+2)}^3=x^3+6x^2+12x+8$

Step-by-step solution

Part a. Step 1. Definition of polynomial.

A polynomial in the variable x is an expression of the form

$a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{.......}+a_{1}x+a_0$where$a_0,a_1.........,a_n$ are real numbers and n is a non-negative integers.

Part a. Step 2. Given expression.

The expression is$3(x+6)+4(2x-5)$

Part a. Step 3. Solving the expression.

$$\begin{gathered} 3(x+6)+4(2x-5) \\ 3x+18+8x-20\left(\mathrm{Multiply}\right) \\ 11x-2\left(\mathrm{Simplify}\right) \end{gathered}$$

Part b. Step 1. Definition of polynomial.

A polynomial in the variable x is an expression of the form

$a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{.......}+a_{1}x+a_0$where$a_0,a_1\mathrm{.........},a_n$ are real numbers and n is a non negative integers.

Part b. Step 2. Given expression.

The expression is$(x+3)(4x-5)$

Part b. Step 3. Solving the expression.

$$\begin{gathered} (x+3)(4x-5) \\ 4x^2-5x+12x-15\left(\mathrm{Multiply}\right) \\ 4x^2+7x-15\left(\mathrm{Simplify}\right) \end{gathered}$$

Part c. Step 1. Definition of polynomial.

A polynomial in the variable x is an expression of the form

$a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{.......}+a_{1}x+a_0$where$a_0,a_1\mathrm{.........},a_n$ are real numbers and n is a non negative integers.

Part c. Step 2. Given expression.

The expression is$(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})$

Part c. Step 3. Solving the expression.

$$\begin{gathered} (\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b}) \\ \sqrt{a}\sqrt{a}-\sqrt{a}\sqrt{b}+\sqrt{a}\sqrt{b}-\sqrt{b}\sqrt{b}\left(\mathrm{Multiply}\right) \\ a-b\left(\mathrm{Simplify}\right) \end{gathered}$$

Part d. Step 1. Definition of polynomial.

A polynomial in the variable x is an expression of the form

$a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{.......}+a_{1}x+a_0$where$a_0,a_1\mathrm{.........},a_n$ are real numbers and n is a non-negative integers.

Part d. Step 2. Given expression.

The expression is${(2x+3)}^2$

Part d. Step 3. Solving the expression.

$$\begin{gathered} {(2x+3)}^2=(2x+3)(2x+3) \\ =4x^2+6x+6x+9\left(\mathrm{Multiply}\right) \\ =4x^2+12x+9\left(\mathrm{Simplify}\right) \end{gathered}$$

Part e. Step 1. Definition of polynomial.

A polynomial in the variable x is an expression of the form

$a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{.......}+a_{1}x+a_0$where$a_0,a_1\mathrm{.........},a_n$ are real numbers and n is a non-negative integers.

Part e. Step 2. Given expression.

The expression is${(x+2)}^3$

Part e. Step 3. Solving the expression.

$$\begin{gathered} {(x+2)}^3=(x+2)(x+2)(x+2) \\ =(x^2+4+4x)(x+2)\left(\mathrm{Multiply}\right) \\ =({\mathrm{x}}^3+2{\mathrm{x}}^2++4\mathrm{x}+8+4{\mathrm{x}}^2+8\mathrm{x})\left(\mathrm{Multiply}\right) \\ ={\mathrm{x}}^3+6{\mathrm{x}}^2+12\mathrm{x}+8\left(\mathrm{Simplify}\right) \end{gathered}$$